BIOLOGICAL RHYTHMS AS TEMPORAL
DISSIPATIVE STRUCTURES
ALBERT GOLDBETER
Service de Chimie Physique et Biologie Théorique, Faculté des Sciences,
Université Libre de Bruxelles, B-1050 Brussels, Belgium
CONTENTS
I.
II.
III.
IV.
V.
Introduction
Dissipative Structures in Time and Space
Glycolytic Oscillations
Calcium Oscillations
Pulsatile Intercellular Communication in Dictyostelium
A. Oscillations of cAMP
B. Link with Pulsatile Hormone Secretion
VI. Circadian Rhythms
A. Circadian Rhythms in Drosophila
B. The Mammalian Circadian Clock
C. Link with Disorders of the Sleep–Wake Cycle
D. Long-Term Suppression of Circadian Rhythms by a Single Light Pulse
E. Stochastic Versus Deterministic Models for Circadian Rhythms
VII. The Cell-Cycle Clock
VIII. Newly Discovered Cellular Rhythms
A. Oscillations of p53 and NF-KB
B. Segmentation Clock
C. Nucleocytoplasmic Oscillations of the Transcription Factor Msn2 in Yeast
IX. Complex Oscillatory Behavior
X. Concluding Remarks
Acknowledgments
References
Special Volume in Memory of Ilya Prigogine: Advances in Chemical Physics, Volume 135,
edited by Stuart A. Rice
Copyright # 2007 John Wiley & Sons, Inc.
253
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I.
INTRODUCTION
From the very beginning of his scientific path, which spanned more than six
decades, Ilya Prigogine was attracted by the question of how order in time and
space spontaneously arises in chemical and biological systems. The title of an
article he published in 1969, ‘‘Structure, Dissipation and Life,’’ reflects this
theme, which long remained a central preoccupation in his research. In this
chapter I will show how the views of Ilya Prigogine on nonequilibrium selforganization found multifarious applications in the life sciences. I will focus on
temporal self-organization in the form of oscillatory behavior, which is
ubiquitous in biological systems. One question that naturally arises is, Why
are there so many biological rhythms?
Until the 1950s, the rare periodic phenomena known in chemistry, such as the
reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions
were also known in electrochemistry. The link was made between the cardiac
rhythm and electrical oscillators [2]. New examples of oscillatory chemical
reactions were later discovered [3, 4]. From a theoretical point of view, the first
kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar
equations were proposed soon after by Volterra [6] to account for oscillations in
predator–prey systems in ecology. The next important advance on biological
oscillations came from the experimental and theoretical studies of Hodgkin and
Huxley [7], which clarified the physicochemical bases of the action potential in
electrically excitable cells. The theory that they developed was later applied [8] to
account for sustained oscillations of the membrane potential in these cells.
Remarkably, the classic study by Hodgkin and Huxley appeared in the same year
as Turing’s pioneering analysis of spatial patterns in chemical systems [9].
The approach of periodic phenomena in physicochemical terms made further
progress when Prigogine and Balescu [10] showed that sustained oscillations
can occur far from thermodynamic equilibrium in open chemical systems
governed by appropriate, nonlinear kinetic laws. The model analyzed in that
study was a chemical analogue of the Lotka–Volterra system for predator–prey
oscillations in ecology. The results were later extended by the analysis of
abstract models of oscillatory reactions, such as the Brusselator, whose name
was given by Tyson [11] to a theoretical model studied in detail in Brussels by
Lefever, Nicolis, and Prigogine [12]. As shown by these studies, chemical
oscillations can occur at a critical distance from equilibrium, around a steady
state that has become unstable owing to the presence of autocatalytic steps in
the reaction kinetics [12–16].
In the phase space formed by the concentrations of the chemical variables
involved in the reaction, sustained oscillations correspond to the evolution
towards a closed curve called a limit cycle [17]. The time taken to travel once
along the closed curve represents the period of the oscillations. When a single
biological rhythms as temporal dissipative structures
255
limit cycle exists, the system always evolves towards the same closed curve
characterized by a fixed amplitude and period, for a given set of parameter
values, regardless of the initial conditions. It is in this sense that oscillations of
the limit cycle type differ from Lotka–Volterra oscillations, for which an infinity
of closed curves, corresponding to oscillations of different periods and
amplitudes, surround the steady state in the phase space. Then the choice of
any one of the closed trajectories depends on the initial conditions [15, 17].
The developments of the Thermodynamics of Irreversible Processes in the
nonlinear domain permitted Prigogine to place periodic phenomena within the
field of nonequilibrium processes of self-organization [13, 14, 18]. Much as
spatial structures arise in chemical systems beyond a critical point of instability
with respect to diffusion [9], rhythms correspond to a temporal organization that
appears beyond a critical point of instability of a nonequilibrium steady state.
These two types of nonequilibrium self-organization represent dissipative
structures [13, 14] that can be maintained only by the energy dissipation
associated with the exchange of matter between the chemical system and its
environment. Sustained oscillations of the limit cycle type can thus be viewed as
temporal dissipative structures [13, 14, 15, 18]. When it occurs in constant
environmental conditions, periodic behavior provides the clearest sign that a
chemical or biological system operates beyond a point of nonequilibrium
instability. Endogenous rhythms, produced by a system and not by its
environment, are indeed the signature of an instability.
From a mathematical point of view, the onset of sustained oscillations generally
corresponds to the passage through a Hopf bifurcation point [19]: For a critical
value of a control parameter, the steady state becomes unstable as a focus. Before
the bifurcation point, the system displays damped oscillations and eventually
reaches the steady state, which is a stable focus. Beyond the bifurcation point, a
stable solution arises in the form of a small-amplitude limit cycle surrounding the
unstable steady state [15, 17]. By reason of their stability or regularity, most
biological rhythms correspond to oscillations of the limit cycle type rather than to
Lotka–Volterra oscillations. Such is the case for the periodic phenomena in
biochemical and cellular systems discussed in this chapter. The phase plane
analysis of two-variable models indicates that the oscillatory dynamics of neurons
also corresponds to the evolution toward a limit cycle [20]. A similar evolution is
predicted [21] by models for predator–prey interactions in ecology.
The 1970s saw an explosion of theoretical and experimental studies devoted
to oscillating reactions. This domain continues to expand as more and more
complex phenomena are observed in the experiments or predicted theoretically.
The initial impetus for the study of oscillations owes much to the concomitance
of several factors. The discovery of temporal and spatiotemporal organization in
the Belousov–Zhabotinsky reaction [22], which has remained the most
important example of a chemical reaction giving rise to oscillations and waves,
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albert goldbeter
and the elucidation of its reaction mechanism [23] occurred at a time when
thermodynamic advances were establishing the theoretical bases of temporal
and spatial self-organization in chemical systems under nonequilibrium
conditions [10, 13–15, 18].
At the same time as the Belousov–Zhabotinsky reaction provided a chemical
prototype for oscillatory behavior, the first experimental studies on the reaction
catalyzed by peroxidase [24] and on the glycolytic system in yeast (to be
discussed in Section III) demonstrated the occurrence of biochemical
oscillations in vitro. These advances opened the way to the study of the
molecular bases of oscillations in biological systems.
Oscillations represent one of the most striking manifestations of dynamic
behavior in biological systems. In 1936, Fessard [25] published a book entitled
Rhythmic Properties of Living Matter. This book was solely devoted to the
oscillatory properties of nerve cells. It has now become clear that rhythms are
encountered at all levels of biological organization, with periods ranging from a
fraction of a second to years, spanning more than 10 orders of magnitude. The
main types of biological rhythms are listed in Table I, where they are ordered
according to their period, from the fastest rhythms in nerve and muscle cells to
the rhythms of longest period observed in ecology and for the flowering of some
plant species.
New examples of cellular rhythms have recently been uncovered (Table II).
These include periodic changes in the intracellular concentration of the
transcription factor NF-KB and of the tumor suppressors p53, stress-induced
oscillations in the transport of the transcription factor Msn2 between cytoplasm
and nucleus in yeast, the segmentation clock that is responsible for the
TABLE I
Main Biological Rhythms
Biological Rhythm
Neural rhythmsa
Cardiac rhythma
Calcium oscillationsa
Biochemical oscillationsa
Mitotic oscillatora
Hormonal rhythmsa
Circadian rhythmsa
Ovarian cycle
Annual rhythms
Rhythms in ecology and epidemiology
a
Period
0.001 s to 10 s
1s
sec to min
30 s to 20 min
10 min to 24 h
10 min to 3–5 h (24 h)
24 h
28 days (human)
1 year
years
These rhythms can already occur at the cellular level.
Source: Goldbeter [31].
biological rhythms as temporal dissipative structures
257
TABLE II
Some Recently Discovered Cellular Rhythmsa
Cellular Rhythm
Period
Segmentation clock
NFkB
P53
Msn2 in yeast
Yeast transcriptome
90 min
3h
3h
6 min
40 min
a
See section VIII for details.
formation of somites in vertebrates, and whole genome oscillations in yeast.
Some synthetic oscillatory gene circuits were recently constructed, as
exemplified by the Repressilator [26]. Given the rapidly rising interest in the
dynamic behavior of genetic circuits, it is likely that additional examples of
cellular rhythms will be found in a near future.
II.
DISSIPATIVE STRUCTURES IN TIME AND SPACE
In the course of time open systems that exchange matter and energy with their
environment generally reach a stable steady state. However, as shown by
Glansdorff and Prigogine, once the system operates sufficiently far from
equilibrium and when its kinetics acquire a nonlinear nature, the steady state
may become unstable [15, 18]. Feedback regulatory processes and cooperativity
are two major sources of nonlinearity that favor the occurrence of instabilities in
biological systems.
Some of the main types of cellular regulation associated with rhythmic
behavior are listed in Table III. Regulation of ion channels gives rise to the
periodic variation of the membrane potential in nerve and cardiac cells [27, 28;
for a recent review of neural rhythms see, for example, Ref. 29]. Regulation of
enzyme activity is associated with metabolic oscillations, such as those that
occur in glycolysis in yeast and muscle cells. Calcium oscillations originate
TABLE III
Biological Regulations and Examples of Associated Cellular Rhythms
Regulation of
Ion channel
Enzyme
Receptor
Transport
Gene expression
Examples of Associated Rhythms
Neural and cardiac rhythms
Glycolytic oscillations in yeast
cAMP oscillations in Dictyostelium
Ca2þ oscillations
Circadian rhythms, segmentation clock
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albert goldbeter
from the control of transport processes within the cell. Regulation of receptors,
coupled to the regulation of enzyme activity, can give rise to periodic behavior,
as exemplified by oscillations of cyclic AMP (cAMP) in Dictyostelium cells.
Regulation of gene expression represents a key type of cellular regulation
involved in the mechanism of circadian rhythms and of the segmentation clock.
When the steady state becomes unstable, the system moves away from it and
often undergoes sustained oscillations around the unstable steady state. In the phase
space defined by the system’s variables, sustained oscillations generally
correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a
limit cycle is not the only possible behavior when a steady state becomes unstable
in a spatially homogeneous system. The system may evolve toward another stable
steady state—when such a state exists. The most common case of multiple steady
states, referred to as bistability, is of two stable steady states separated by an
unstable one. This phenomenon is thought to play a role in differentiation [30].
When spatial inhomogeneities develop, instabilities may lead to the emergence of
spatial or spatiotemporal dissipative structures [15]. These can take the form of
propagating concentration waves, which are closely related to oscillations.
3.5
3
per mRNA, M
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
Total PER protein, Pt
Figure 1. In most examples of biological rhythms, sustained oscillations correspond to the
evolution toward a limit cycle. The limit cycle shown here was obtained in a model for circadian
oscillations of the PER protein and per mRNA in Drosophila [107].
biological rhythms as temporal dissipative structures
259
Elucidating the molecular mechanism of a biological rhythm largely reduces
to identifying the feedback processes that lie at the core of the oscillations. The
latter may originate from positive or negative feedback, or from a mixture of
both. The interplay between a large number of variables coupled through
multiple regulatory interactions makes it difficult, if not impossible, to fully
grasp the dynamics of oscillatory behavior without resorting to modeling and
computer simulations [31, 32].
As indicated above, theoretical models for biological rhythms were first used
in ecology to study the oscillations resulting from interactions between
populations of predators and preys [6]. Neural rhythms represent another field
where such models were used at an early stage: The formalism developed by
Hodgkin and Huxley [7] still forms the core of most models for oscillations of
the membrane potential in nerve and cardiac cells [33–35]. Models were
subsequently proposed for oscillations that arise at the cellular level from
regulation of enzyme, receptor, or gene activity (see Ref. 31 for a detailed list of
references).
Some of the main examples of biological rhythms of nonelectrical nature are
discussed below, among which are glycolytic oscillations (Section III),
oscillations and waves of cytosolic Ca2þ (Section IV), cAMP oscillations that
underlie pulsatile intercellular communication in Dictyostelium amoebae
(Section V), circadian rhythms (Section VI), and the cell cycle clock
(Section VII). Section VIII is devoted to some recently discovered cellular
rhythms. The transition from simple periodic behavior to complex oscillations
including bursting and chaos is briefly dealt with in Section IX. Concluding
remarks are presented in Section X.
III.
GLYCOLYTIC OSCILLATIONS
Glycolytic oscillations in yeast cells provided one of the first examples of
oscillatory behavior in a biochemical system. They continue to serve as a
prototype for cellular rhythms. This oscillatory phenomenon, discovered some
40 years ago [36, 37] and still vigorously investigated today [38], was important
in several respects: First, it illustrated the occurrence of periodic behavior in a
key metabolic pathway. Second, because they were soon observed in cell
extracts, glycolytic oscillations provided an instance of a biochemical clock
amenable to in vitro studies. Initially observed in yeast cells and extracts,
glycolytic oscillations were later observed in muscle cells and evidence exists for
their occurrence in pancreatic b-cells in which they could underlie the pulsatile
secretion of insulin [39].
The molecular mechanism of glycolytic oscillations has been discussed for
long [31, 38, 40–42]. Because glycolysis represents a system of enzymatic
reactions coupled through different intermediates such as ATP and NADH,
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albert goldbeter
which impinge on multiple steps in the pathway, it is difficult to isolate a single
enzymatic step that would be responsible for oscillatory behavior. However,
there is a large, if not unanimous, consensus in attributing to the enzyme
phosphofructokinase (PFK) a prominent role in the instability-generating
mechanism that leads to glycolytic oscillations. This role, recognized since the
early experimental studies on the phenomenon, is due to the peculiar regulation
of PFK, which is activated by a reaction product, ADP. Such product activation
means that the PFK reaction is autocatalytic, a feature long shown to be
associated with nonequilibrium instabilities [15, 18]. Self-amplification of PFK
due to product activation of the enzyme was at the core of early models
proposed for glycolytic oscillations [43–45].
A two-variable model taking into account the allosteric (i.e. cooperative)
nature of the enzyme and the autocatalytic regulation exerted by the product
shows the occurrence of sustained oscillations. Beyond a critical parameter
value, the steady state admitted by the system becomes unstable and the system
evolves toward a stable limit cycle corresponding to periodic behavior. The
model accounts for most experimental data, particularly the existence of a
domain of substrate injection rates producing sustained oscillations, bounded by
two critical values of this control parameter, and the decrease in period observed
when the substrate input rate increases [31, 45, 46].
Whereas two bifurcation values for the glucose input rate define the domain
of oscillations in yeast extracts [40], only a single bifurcation value below which
oscillations occur is found in intact yeast cells [47]. This does not necessarily
imply a difference in oscillatory mechanism but merely indicates that in intact
cells the glucose transporter becomes saturated before the intracellular glucose
input has reached the upper bifurcation value above which oscillations disappear
in yeast extracts [38].
If the primary role of PFK in generating glycolytic oscillations has long been
stressed and substantiated by models based on its regulatory properties, other
reactions of the glycolytic pathways are coupled to PFK and may thus influence
its dynamic behavior. More complex models incorporating a large number of
enzymatic reactions and of glycolytic intermediates have been proposed. This
alternative approach to modeling was pioneered more than four decades ago by
Garfinkel and Hess [48], who early on presented a comprehensive computer
model for the glycolytic pathway. This work represents one of the first studies
in a field currently known as (computational) systems biology. Other full-scale
models of the yeast glycolytic system were subsequently proposed [49, 50].
The question of how glycolytic oscillations synchronize in a population of
yeast cells is of great current interest [51]. It has long been known that the
oscillations disappear in a yeast suspension when the cell density decreases
below a critical value. Acetaldehyde appears to act as synchronizing factor in
such suspensions [52], and the way it allows cells to synchronize is being
biological rhythms as temporal dissipative structures
261
studied in both an experimental and theoretical manner. The link between
glycolytic oscillations and the pulsatile secretion of insulin in pancreatic b cells
[53] is another topic of current concern. Models for the latter phenomenon rely
on the coupling between intracellular metabolic oscillations and an ionic
mechanism generating action potentials. Such coupling results in bursting
oscillations of the membrane potential, which are known to accompany insulin
secretion in these cells [54, 55].
IV.
CALCIUM OSCILLATIONS
The three best-known examples of biochemical oscillations were found during
the decade 1965–1975 [40, 41]. These include the peroxidase reaction, glycolytic
oscillations in yeast and muscle, and the pulsatile release of cAMP signals in
Dictyostelium amoebae (see Section V). Another decade passed before the
development of Ca2þ fluorescent probes led to the discovery of oscillations in
intracellular Ca2þ. Oscillations in cytosolic Ca2þ have since been found in a
variety of cells where they can arise spontaneously, or after stimulation by
hormones or neurotransmitters. Their period can range from seconds to minutes,
depending on the cell type [56]. The oscillations are often accompanied by
propagation of intracellular or intercellular Ca2þ waves. The importance of Ca2þ
oscillations and waves stems from the major role played by this ion in the control
of many key cellular processes—for example, gene expression or neurotransmitter secretion.
In cells that use Ca2þ as second messenger, binding of an external signal to a
cell membrane receptor activates phospholipase C (PLC), which, in turn,
synthesizes inositol 1,4,5-trisphosphate (InsP3). This metabolite binds to an
InsP3 receptor located on the membrane of internal Ca2þ stores (endoplasmic or
sarcoplasmic reticulum) and thereby triggers the release of Ca2þ into the
cytoplasm of the cell [56]. A conspicuous feature of Ca2þ release is that it is
self-amplified: Cytosolic Ca2þ triggers the release of Ca2þ from intracellular
stores into the cytosol, a process known as Ca2þ -induced Ca2þ release (CICR)
[57, 58].
A first model for cytosolic Ca2þ oscillations was based on the activation of
PLC by Ca2þ [59]. Although this positive feedback has been observed in some
cell types, CICR represents a more general self-amplifying process underlying the
oscillations. Several processes limit the explosive nature of self-amplification.
A simple two-variable model for signal-induced Ca2þ oscillations based on
CICR accounts for oscillations of cytosolic Ca2þ [60]. Sustained oscillations
occur between two critical values of the stimulus intensity—for example, two
critical levels of an extracellular hormonal signal (see Fig. 2). Below the lower
critical value, a low steady-state level of cytosolic Ca2þ is established; above the
larger critical value, the system evolves toward a higher, stable steady-state level
262
Cytosolic Ca 2+
albert goldbeter
Max cytosolic Ca2+
High steady state
of cytosolic Ca2+
Unstable steady state
Low steady state
of cytosolic Ca2+
Min cytosolic Ca2+
βc1
βc2
Degree of cell stimulation, β
Figure 2. Schematic bifurcation diagram showing the domain and amplitude of intracellular
Ca2þ oscillations as a function of the degree of external stimulation, b, which is used as control
parameter. Sustained Ca2þ oscillations occur in a range of stimulation between the two critical b
values denoted bc1 and bc2. The maximum and minimum of cytosolic Ca2þ oscillations are plotted
as a function of b in this range, in which the dashed line refers to the unstable steady state. On the
left and right sides of the oscillatory domain, the system evolves to a stable steady state (solid line)
corresponding to a low and high level of cytosolic Ca2þ , respectively. The bifurcation diagram is
obtained in a two-variable model for Ca2þ oscillations based on CICR (see Goldbeter et al. [60] for a
nonschematic version of the diagram). A similar bifurcation diagram with a domain of sustained
oscillations bounded by two bifurcation values of the control parameter is obtained for glycolytic
oscillations as a function of the substrate injection rate in yeast extracts [46, 193]. In intact yeast
cells, however, the upper bifurcation point cannot be reached, likely because, as described in
Section III, the glucose transporter is saturated before the bifurcation value for the substrate input is
reached inside the cell.
of cytosolic Ca2þ . The model predicts that the frequency of Ca2þ oscillations
rises with the degree of stimulation, as observed experimentally. In this minimal
model the level of intracellular InsP3 is treated as a control parameter reflecting
the degree of external stimulation. More complex models for Ca2þ oscillations
are based on more detailed descriptions of InsP3 receptor kinetics [61; for a
recent review see Ref. 62] but still attribute to CICR a primary role in the origin
of repetitive Ca2þ spiking.
Mathematical models for Ca2þ signaling were subsequently developed in
two additional directions. First, waves of intra- or intercellular Ca2þ can be
modeled by incorporating the diffusion of cytosolic Ca2þ or the passage of Ca2þ
or InsP3 from cell to cell through gap junctions [62–65]. While most models for
biological rhythms as temporal dissipative structures
263
Ca2þ waves are deterministic, stochastic simulations were used to clarify the
nature of local increases of cytosolic Ca2þ known as blips or puffs which are
thought to trigger the onset of waves [56, 66]. Second, models are used to probe
mechanisms for encoding Ca2þ spikes in terms of their frequency. A variety of
physiological responses are controlled by the frequency and waveform of Ca2þ
oscillations, such as gene expression during development [67]. Among the
processes that could underlie such frequency encoding are protein (de)phosphorylation by a Ca2þ -dependent kinase (phosphatase) [60], or the Ca2þ dependence of calmodulin-kinase II [68, 69]. A study combining experimental
and modeling approaches showed the possibility of frequency encoding of Ca2þ
spikes by interplay with cyclic AMP signaling [70].
The characteristics of cytosolic Ca2þ oscillations vary from one cell type to
another. One source for this variability is the existence of three isoforms of the
inositol 1,4,5-trisphosphate receptor, InsP3R, whose proportions vary in
different cells. Upon binding of InsP3, the InsP3R functions as a Ca2þ channel
on the endoplasmic reticulum, allowing passage of Ca2þ into the cytoplasm of
the cell. The relative amounts of each isoform of the InsP3 receptor affect the
time course of Ca2þ changes after agonist stimulation, because the effect of
Ca2þ on the three isoforms are different. The different modes of Ca2þ
oscillatory behavior have recently been modeled as a function of the relative
proportions of the three InsP3R isoforms [71]. Based on a comparative study of
various models, Sneyd et al. [72] recently proposed a method for determining
the dependence of Ca2þ oscillations on InsP3 oscillations, to determine whether
InsP3 plays an active role in the oscillatory mechanism or passively follows the
periodic spikes in Ca2þ .
The role of Ca2þ oscillations in some physiological disorders begins to be
characterized. Two recent studies provide examples of how changes in Ca2þ
oscillatory signaling possess profound implications for developmental processes. First, Beltramello et al. [73] showed that a mutation associated with
hereditary deafness reduces metabolic coupling mediated by InsP3 and impairs
the propagation of intercellular Ca2þ waves. Second, Uhlen et al. [74] recently
demonstrated that the Noonan syndrome, a human developmental disorder often
accompanied by congenital heart abnormalities, is caused by alterations in the
Ca2þ oscillatory control of the transcription factor NFAT. These examples
illustrate the impact of changes in the normal patterns of oscillations and waves
of Ca2þ on the pathogenesis of some inherited human diseases.
V.
PULSATILE INTERCELLULAR COMMUNICATION
IN DICTYOSTELIUM
While intracellular information can be encoded in the frequency of signalinduced Ca2þ spikes, some extracellular signals can themselves be produced in a
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albert goldbeter
periodic, pulsatile manner. Examples of pulsatile intercellular communication
include episodic hormone secretion and pulsatile signals of cAMP in the slime
mold Dictyostelium discoideum. The latter phenomenon represents a prototype
both for spatiotemporal self-organization and for pulsatile signaling in
intercellular communication [31].
A.
Oscillations of cAMP
After starvation, Dictyostelium amoebae undergo a transition from a unicellular
to a multicellular phase of their life cycle. By a chemotactic response to cAMP
signals, up to 105 amoebae collect around cells behaving as aggregation centers.
These centers release cAMP with a period of about 5 min; surrounding cells relay
the chemotactic signal toward the periphery of the aggregation field. Relay and
oscillations of cAMP result in the formation of concentric or spiral waves of
aggregating cells [75].
Models help to clarify the mechanism of cAMP oscillations in Dictyostelium
[76, 77]. The mechanism involves both positive and negative feedback. Binding
of extracellular cAMP to a cell surface receptor leads to the activation of
adenylate cyclase, which catalyzes the synthesis of intracellular cAMP.
Transport of cAMP into the extracellular medium creates a positive feedback
loop, which elicits a rapid rise in cAMP synthesis. For sustained oscillations to
occur, this rise in cAMP must be self-limiting, so that cAMP first levels off
before decreasing to its minimum level. Models confirm that negative feedback
due to cAMP-induced receptor desensitization through reversible phosphorylation can play such a role in limiting self-amplification [76]. Once the levels of
intra- and extracellular cAMP are sufficiently low, dephosphorylation can
resensitize the receptor. The ensuing buildup of extracellular cAMP progressively brings it to the threshold above which self-amplification triggers a
new pulse.
Numerical simulations indicate that relay of cAMP pulses represents a
different mode of dynamic behavior, closely related to oscillations. Just before
autonomous oscillations break out, cells in a stable steady state can amplify
suprathreshold variations in extracellular cAMP in a pulsatory manner. Thus,
relay and oscillations of cAMP are produced by a unique mechanism in adjacent
domains in parameter space. The two types of dynamic behavior are analogous
to the excitable or pacemaker behavior of nerve cells.
Theoretical models shed light on additional aspects of pulsatile cAMP
signaling in Dictyostelium. First, like Ca2þ spikes, cAMP pulses are frequency
encoded. Only pulses delivered at 5-min intervals are capable of accelerating
slime mold development after starvation. Simulations indicate that frequency
encoding is based on reversible receptor desensitization [76]. The kinetics
of receptor resensitization dictates the interval between successive pulses
required for a maximum relay response [78]. Second, cAMP oscillations in
biological rhythms as temporal dissipative structures
265
Dictyostelium provide a prototype for the ontogenesis of biological rhythms.
The amoebae become capable of relaying extracellular cAMP pulses only a few
hours after the beginning of starvation, before acquiring the property of
autonomous oscillations. Models show that these developmental transitions can
be brought about by the continuous increase in certain biochemical parameters
such as the activities of adenylate cyclase or phosphodiesterase, the enzyme
that degrades cAMP. In parameter space, these biochemical changes define
a developmental path that successively crosses domains corresponding to
different types of dynamic behavior, from no relay to relay, and finally to
oscillations [31, 79].
Models are also being used to probe the mechanisms underlying the
formation of concentric or spiral waves of cAMP responsible for the
spatiotemporal patterns observed during aggregation [80]. Among the factors
shown to play a role in the transition between the two types of waves are the
activity of extracellular phosphodiesterase [81] and desynchronization of the
cells that follow the developmental path after starvation [82]. The model based
on the positive feedback mechanism coupled to receptor desensitization also
accounts for the propagation of planar and scroll waves within the multicellular
slug formed by the amoebae after aggregation [83].
In recent years, work by Loomis and co-workers has raised the possibility
that cAMP oscillations in D. discoideum may originate from an intracellular
regulatory network rather than from the mixed positive and negative feedback
exerted by extracellular cAMP [84, 85]. These authors obtained evidence for an
intracellular feedback loop involving MAP kinase and the cAMP-dependent
protein kinase, PKA. The later enzyme would inactivate adenylate cyclase after
a cAMP pulse. Numerical simulations of a model based on this intracellular
negative feedback loop confirm that it can produce sustained oscillations of
cAMP.
To establish which of the two feedback loops plays a prominent role in the
origin of cAMP oscillations, Cox and co-workers recently examined the
patterns of wavelike aggregation in a variety of mutants lacking components of
the intracellular and extracellular regulatory loops. They reached the
conclusion that the primary (but not necessarily sole) source of the oscillations
resides in the regulation exerted by extracellular cAMP upon binding to its
membrane receptor [86]. Interestingly, the possibility of cAMP oscillations
due to intracellular regulation of adenylate cyclase by PKA seems to exist not
only in Dictyostelium but also in yeast. In this organism, Jacquet et al. [87]
recently observed a stress-induced oscillatory shuttling of the transcription
factor Msn2 between cytosol and nucleus. They since obtained evidence
suggesting that this periodic phenomenon is caused by intracellular cAMP
oscillations, via the control of adenylate cyclase by PKA (see Section VIII). In
this view, periodic activation of PKA by cAMP oscillations in yeast would
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underlie the repetitive, coherent shuttling of the transcription factor Msn2 into
and out of the nucleus.
B.
Link with Pulsatile Hormone Secretion
Pulsatile cAMP signaling in Dictyostelium is closely related with pulsatile
hormone secretion in higher organisms. It is now clear that most hormones
are secreted in a pulsatile rather than continuous manner [88] and that the
temporal pattern of a hormone is often as important as its concentration in the
blood [89]. The best examples of pulsatile hormone secretion are the
gonadotropin-releasing hormone (GnRH) released by the hypothalamus with
a periodicity of 1 h in man and rhesus monkey [90], the growth hormone
(GH) secreted by the hypothalamus with a period of 3–5 h [91], and insulin
secreted by pancreatic b cells with a period close to 13 min in man [53]. In
the cases of GnRH and GH—the effect is less clear-cut for insulin—the
frequency of the pulses governs the physiological efficacy of hormone
stimulation [90, 91].
A general model for a two-state receptor subjected to periodic ligand
variations shows that frequency encoding of hormone pulses may rely on
reversible desensitization in target cells, as in the case of cAMP pulses in
Dictyostelium [78, 92]. The mechanism of the hypothalamic GnRH pulse
generator is still unknown and provides an important challenge for both
experiments and theory. The basis of pulsatile GH secretion has been studied by
a modeling approach [93]. In b cells, pulsatile insulin release could originate
from insulin feedback on glucose transport into the cells [94] or from oscillatory
membrane activity driven by glycolytic oscillations [53–55]. Together with such
metabolic oscillations, membrane potential bursting and Ca2þ oscillations in b
cells illustrate the multiplicity of rhythms that can be encountered in a given
cell type.
VI.
CIRCADIAN RHYTHMS
The most ubiquitous biological rhythms are those that occur with a period
close to 24 h in all eukaryotes and in some prokaryotes such as
cyanobacteria. These circadian rhythms allow organisms to adapt to the
natural periodicity of the terrestrial environment, which is characterized by
the alternation of day and night due to rotation of the earth on its axis.
Circadian clocks provide cells with an endogenous mechanism, allowing
them to anticipate the time of day.
Experimental advances during the last decade have clarified the molecular
bases of circadian rhythms, first in Drosophila and Neurospora, and more
recently in cyanobacteria, plants, and mammals [95–99]. In nearly all cases
investigated so far, it appears that circadian rhythms originate from the negative
biological rhythms as temporal dissipative structures
267
feedback exerted by a protein on the expression of its gene [100]. Circadian
rhythms in cyanobacteria appear to be based on a different molecular
mechanism, which can be uncoupled from transcriptional control. Thus the
circadian oscillation in the phosphorylation of the cyanobacterial KaiC clock
protein has recently been reconstituted in vitro [101].
Before details on the molecular mechanism of circadian rhythms began to be
uncovered, theoretical models borrowed from physics were used to investigate
the dynamic properties of circadian clocks. The relative simplicity of these
models explains why their use continues to this day. Thus, the Van der Pol
equations, derived for an electrical oscillator, served for modeling the response
of human circadian oscillations to light [102] and to account for experimental
observations on increased fitness due to resonance of the circadian clock with
the external light–dark (LD) cycle in cyanobacteria [103, 104]. The earliest
model predicting oscillations due to negative feedback was proposed by
Goodwin [105], at a time when the role played by such a regulatory mechanism
in the origin of circadian rhythms was not yet known. Models based on
Goodwin’s equations are still being used in studies of circadian oscillations—
for example, in Neurospora [106].
A.
Circadian Rhythms in Drosophila
Molecular models for circadian rhythms were initially proposed [107] for
circadian oscillations of the PER protein and its mRNA in Drosophila, the first
organism for which detailed information on the oscillatory mechanism became
available [100]. The case of circadian rhythms in Drosophila illustrates how the
need to incorporate experimental advances leads to a progressive increase in the
complexity of theoretical models. A first model governed by a set of five kinetic
equations is shown in Fig. 3A; it is based on the negative control exerted by the
PER protein on the expression of the per gene [107]. Numerical simulations
show that for appropriate parameter values, the steady state becomes unstable
and limit cycle oscillations appear (Fig. 1).
The early model based on PER alone did not account for the effect of light on
the circadian system. Experiments subsequently showed that a second protein,
TIM, forms a complex with PER and that light acts by inducing TIM
degradation [96]. An extended, 10-variable model was then proposed [108], in
which the negative regulation is exerted by the PER–TIM complex (Fig. 3B).
This model produces essentially the same result, sustained oscillations in
continuous darkness. In addition, it accounts for the behavior of mutants and
explicitly incorporates the effect of light on the TIM degradation rate. Thereby
the model can account for the entrainment of the oscillations by light-dark (LD)
cycles and for the phase shifts induced by light pulses.
Subsequent experimental studies have shown that the mechanism of
circadian rhythms in Drosophila is more complex, since the negative
268
albert goldbeter
Figure 3. Molecular models of increasing complexity considered for circadian oscillations.
(A) Model for circadian oscillations in Drosophila based on negative autoregulation of the per gene
by its protein product PER [31, 107]. The model incorporates gene transcription into per mRNA,
transport of per mRNA into the cytosol as well as mRNA degradation, synthesis of the PER protein
at a rate proportional to the per mRNA level, reversible phosphorylation and degradation of PER,
and transport of PER into the nucleus where it represses the transcription of the per gene. The model
is described by a set of five kinetic equations. (B) Model for circadian oscillations in Drosophila
incorporating the formation of a complex between the PER and TIM proteins [108]. The model is
described by a set of 10 kinetic equations. (C) Model for circadian oscillations in mammals
incorporating indirect, negative autoregulation of the Per and Cry genes through binding of the PERCRY dimer to the complex formed between the two activating proteins CLOCK and BMAL1. Also
considered is the negative feedback exerted by the latter proteins on the expression of their genes.
Synthesis, reversible phosphorylation, and degradation of the various proteins are taken into account.
The model is described by a set of 16 kinetic equations, or 19 when the Rev-Erba gene is
incorporated into the model [114]. For appropriate parameter values, all three models admit
sustained circadian oscillations in conditions corresponding to continuous darkness. The effect of
light is taken into account in models (B) and (C) by incorporating light-induced TIM degradation or
light-induced Per expression, respectively.
autoregulation exerted by the PER–TIM complex on gene expression is indirect
(see below).
B.
The Mammalian Circadian Clock
The pacemaker generating circadian rhythms in mammals is located in the
suprachiasmatic nuclei of the hypothalamus. Recent studies have shown,
however, that a number of peripheral circadian oscillators operate in tissues
such as liver and heart [109]. Theoretical models for circadian rhythms in
Drosophila bear on the mechanism of circadian oscillations in mammals, where
homologues of the per gene exist and negative autoregulation of gene expression
is also found [96]. However, in mammals, the role of TIM as a partner for PER is
played by the CRY protein, and light acts by inducing gene expression rather
than protein degradation as in Drosophila. A further analogy between
biological rhythms as temporal dissipative structures
269
Figure 3. (Continued)
Drosophila and mammals is that the negative feedback on gene expression is
indirect: The PER-TIM or PER-CRY complexes exert their repressive effect by
binding to a complex of two proteins, CLOCK-CYC or CLOCK-BMAL1 in the
fly [110] and in mammals [111], respectively. These proteins activate per and tim
(or cry) gene expression. Thus negative feedback occurs by counteracting the
effect of gene activators. Additional feedback loops are present, such as the
negative feedback exerted by CLOCK or BMAL1 on the expression of their
genes. These controls are removed upon formation of the complex with the PERTIM or PER-CRY dimers.
Further extensions of the model are required to address the dynamical
consequences of these additional regulatory loops and of the indirect nature of
the negative feedback on gene expression. Such extended models have been
proposed for Drosophila [112, 113] and mammals [113]. The model for the
circadian clock mechanism in mammals is schematized in Fig. 3C. The
presence of additional mRNA and protein species, as well as of multiple
complexes formed between the various clock proteins, complicates the model,
which is now governed by a system of 16 or 19 kinetic equations. Sustained or
damped oscillations can occur in this model for parameter values corresponding
to continuous darkness. As observed in the experiments on the mammalian
clock, Bmal1 mRNA oscillates in opposite phase with respect to Per and Cry
mRNAs [97]. The model displays the property of entrainment by the LD cycle
270
transcription
+
-
transcription
+
+
transcription
Light
mRNA
mRNA
mRNA
BMAL1 nu c-P
BMAL1 cyto-P
Figure 3. (Continued)
BMAL1 nuc
PER-CRY cyto
PER-CRY cyto-P
BMAL1 cyto
PER cyto-P
Iε
PER cyto
CRY cyto
CRY cyto-P
CLOCK-BMAL1 nuc
transcription
mRNA
CLOCK cyto
CLOCK nuc
PER-CRY
CLOCK-BMAL1
PER-CRY nuc
PER-CRY nuc-P
biological rhythms as temporal dissipative structures
271
when incorporating the light-induced increase in the rate of Per expression. A
more detailed model containing a much larger number of variables has been
proposed for the mammalian circadian clock [115].
Knowledge of the detailed mechanism underlying circadian rhythms
continues to be refined as new experiments reveal novel facets of the oscillatory
machinery. Thus, a link has recently been established between chromatin
structure and the circadian oscillatory mechanism. The CLOCK protein indeed
functions as a histone acetyltransferase [116]. This enzyme activity is required
for oscillations so that histone modification and the associated chromatin
remodeling are implicated in the origin of circadian rhythmicity.
C.
Link with Disorders of the Sleep–Wake Cycle
The results obtained with the model for the mammalian circadian clock provide
cues for circadian-rhythm-related sleep disorders in humans [117]. Thus
permanent phase shifts in LD conditions could account for (a) the familial
advanced sleep phase syndrome (FASPS) associated with PER hypophosphorylation [118, 119] and (b) the delayed sleep phase syndrome, which is
also related to PER [120]. People affected by FASPS fall asleep around 7:30 P.M.
and awake around 4:30 A.M. The duration of sleep is thus normal, but the phase is
advanced by several hours. Moreover, the autonomous period measured for
circadian rhythms in constant conditions is shorter [121]. The model shows that a
decrease in the activity of the kinase responsible for PER phosphorylation is
indeed accompanied by a reduction of the circadian period in continuous
darkness and by a phase advance upon entrainment of the rhythm by the LD
cycle [114].
For some parameter values the model for the mammalian clock fails to allow
entrainment by 24-h LD cycles, regardless of the amplitude of the light-induced
change in Per expression. The question arises whether there exists a syndrome
corresponding to this mode of dynamic behavior predicted by the model. Indeed
there exists such a syndrome, known as the non-24-h sleep–wake syndrome, in
which the phase of the sleep–wake pattern continuously varies with respect to
the LD cycle; that is, the patient free-runs in LD conditions [117]. Disorders of
the sleep–wake cycle associated with alterations in the dynamics of the
circadian clock belong to the broad class of ‘‘dynamical diseases’’ [122, 123],
although the term ‘‘syndrome’’ seems more appropriate for some of these
conditions.
Another common perturbation of the circadian clock is the jet lag, which
results from an abrupt shift in the phase of the LD cycle to which the rhythm is
naturally entrained. The molecular bases of the jet lag are currently being
investigated [124]. The model for the circadian clock is being used to probe the
various ways by which the clock returns to the limit cycle trajectory after a
sudden shift in the phase of the LD cycle.
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albert goldbeter
D. Long-Term Suppression of Circadian Rhythms by a Single Light Pulse
Circadian rhythms illustrate how theoretical models can provide surprising,
counterintuitive insights. A case in point is the puzzling observation that in some
organisms, circadian rhythms in continuous darkness can be suppressed by a
single pulse of light and restored by a second such pulse. A first theoretical
explanation for this long-term suppression, proposed by Winfree [125], assumes
that the limit cycle in each oscillating cell surrounds an unstable steady state. The
light pulse would act as a critical perturbation that would bring the clock to the
singularity—that is, the steady state. Because the steady state is unstable, each
cell would eventually return to the limit cycle, but with a random phase. The
population of oscillating cells would then be spread out over the entire cycle so
that the cells would be desynchronized and no global rhythm would be
established.
An alternative explanation is based on the coexistence of sustained
oscillations with a stable steady state. Such coexistence has been observed
[126] in the model for circadian rhythms in Drosophila based on negative
autoregulation by the PER-TIM complex (Fig. 3B). In such a situation, the
effect of the light pulse is to bring the clock mechanism into the basin of
attraction of the stable steady state in each oscillating cell, so that the rhythm is
suppressed. A second light pulse then brings the system back to the basin of
attraction of the limit cycle corresponding to circadian oscillations. Without a
model it is impossible to predict the coexistence between a stable steady state
and a stable rhythm. The question remains open as to which one of the two
explanations accounts for long-term suppression of circadian rhythms by a
single light pulse.
E.
Stochastic Versus Deterministic Models for Circadian Rhythms
Only deterministic models for cellular rhythms have been discussed so far. Do
such models remain valid when the numbers of molecules involved are small, as
may occur in cellular conditions? Barkai and Leibler [127] stressed that in the
presence of small amounts of mRNA or protein molecules, the effect of
molecular noise on circadian rhythms may become significant and may
compromise the emergence of coherent periodic oscillations. The way to assess
the influence of molecular noise on circadian rhythms is to resort to stochastic
simulations [127–129]. Stochastic simulations of the models schematized in
Fig. 3A,B show that the dynamic behavior predicted by the corresponding
deterministic equations remains valid as long as the maximum numbers of
mRNA and protein molecules involved in the circadian clock mechanism are of
the order of a few tens and hundreds, respectively [128]. In the presence of
molecular noise, the trajectory in the phase space transforms into a cloud of
points surrounding the deterministic limit cycle.
biological rhythms as temporal dissipative structures
273
Stochastic simulations confirm the existence of bifurcation values of the
control parameters bounding a domain in which sustained oscillations occur.
The effect of noise diminishes as the number of molecules increases. Only when
the maximum numbers of molecules of mRNA and protein become smaller than
a few tens does noise begin to obliterate the circadian rhythm. The robustness of
circadian rhythms with respect to molecular noise is enhanced when the rate of
binding of the repressor molecule to the gene promoter increases [128].
Conditions that enhance the resistance of genetic oscillators to random
fluctuations have been investigated [130].
VII.
THE CELL-CYCLE CLOCK
The cell cycle is a key process that recurs in a periodic manner. Early cell cycles
in amphibian embryos are driven by a mitotic oscillator. This oscillator produces
the repetitive activation of the cyclin-dependent kinase cdk1, also known as cdc2
[131]. Cyclin synthesis is sufficient to drive repetitive cell division cycles in
amphibian embryonic cells [132]. The period of these relatively simple cell
cycles is of the order of 30 min. In somatic cells the cell cycle becomes longer,
with durations of up to 24 h or more, owing to the presence of checkpoints that
ensure that a cell cycle phase is properly completed before the cell progresses to
the next phase. The cell cycle goes successively through the phases G1, S (DNA
replication), G2, and M (mitosis) before a new cycle starts in G1. After mitosis
cells can also enter a quiescent phase G0, from which they enter G1 under
mitogenic stimulation.
Models of reduced complexity have first been proposed for the early cell
cycles in amphibian embryos. These models are based on the activation of the
kinase cdc2 upon binding of cyclin. One of these models predicts that limit
cycle oscillations in cdc2 activity may arise from the activation by cdc2 of
cyclin degradation. Indeed, cdc2 activates the anaphase-promoting complex
(APC), which leads to cyclin destruction and subsequently to cdc2 inactivation.
Such a negative feedback regulation is capable of producing sustained
oscillatory behavior in the presence of thresholds and delays, both of which
are linked and naturally arise in the control of cdc2 by phosphorylationdephosphorylation [31, 133].
Positive feedback is also involved in the control of cdc2 by reversible
phosphorylation. Thus, cdc2 activates the phosphatase cdc25, which catalyzes
the dephosphorylation and concomitant activation of the kinase cdc2. The
model based on negative feedback in cyclin-cdc2 interactions can be extended
to take this positive feedback into account. Sustained oscillations can be
obtained in these conditions [31, 134], but the waveform of cdc2 in the course
of oscillations now displays a plateau. This plateau is due to the occurrence
of a phenomenon of bistability, which is accompanied by hysteresis, as
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albert goldbeter
shown theoretically and experimentally in Xenopus egg extracts by Pomerening
et al. [135] and Sha et al. [136]. The effect of suppressing the positive feedback
loop on the occurrence of cdc2 oscillations was investigated in recent
experiments [137].
The interplay between oscillations and bistability has been addressed in
detailed molecular models for the cell cycles of amphibian embryos, yeast and
somatic cells [138–141]. The predictions of a detailed model for the cell cycle
in yeast were successfully compared with observations of more than a hundred
mutants [142]. Other theoretical studies focus on the dynamical properties of
particular modules of the cell cycle machinery such as that controlling the G1/S
transition [143].
If the cell cycle in amphibian embryonic cells appears to be driven by a limit
cycle oscillator, the question arises as to the precise dynamical nature of more
complex cell cycles in yeast and somatic cells. Novak et al. [144] constructed a
detailed bifurcation diagram for the yeast cell cycle, piecing together the
diagrams obtained as a function of increasing cell mass for the transitions
between the successive phases of the cell cycle. In these studies, cell mass plays
the role of control parameter; a critical mass has to be reached for cell division
to occur, provided that it coincides with a surge in cdk1 activity which triggers
the G2/M transition.
The periodic recurrence of cell division suggests that globally the cell cycle
functions like an autonomous oscillator. An extended model incorporating the
sequential activation of the various cyclin-dependent kinases, followed by their
inactivation, shows that even in the absence of control by cell mass, this
sequence of biochemical events can operate as a limit cycle oscillator [145].
This supports the union of the two views of the cell cycle as dominoes and clock
[146]. Because of the existence of checkpoints, however, the cell cycle stops at
the end of certain phases before engaging in the next one. Thus the cell cycle
looks more like an oscillator that slows down and makes occasional stops. A
metaphor for such behavior is provided by the movement of the round plate on
the table in a Chinese restaurant, which would rotate continuously under the
movement imparted by the participants, were it not for frequent stops.
An alternative approach for modeling the cell cycle considers the sequential
transitions between the G1, S, G2, and M phases without taking into account the
underlying molecular mechanism. Based on a previous study of the dynamics
of hair cycles [147], this phenomenological approach represents the cell cycle as
a stochastic automaton capable of switching between the successive phases,
with a probability related to their duration. The automaton model can reproduce
the distributions between the various phases of the cell cycle at steady state
[148]. This phenomenological model is being used to investigate the effect
of periodic administration of anticancer drugs that interfere with the cell
division cycle.
biological rhythms as temporal dissipative structures
275
Recent experimental studies have uncovered a direct link between the cell
cycle and circadian rhythms. Thus, the circadian clock protein BMAL1 induces
the expression of the gene Wee1, which codes for the protein kinase that
inactivates through phosphorylation the kinase cdk1 that controls the G2/M
transition [149]. This link allows the coupling of cell division to the circadian
clock and explains how the latter may entrain the cell cycle clock in a variety of
cell types.
VIII.
NEWLY DISCOVERED CELLULAR RHYTHMS
In the last decade, and particularly in the last five years, several new examples of
cellular rhythms have been uncovered (see Table II). These include oscillations
in the tumor suppressor p53 and in the transcription factor NF-KB, the
segmentation clock that controls the formation of somites in vertebrates, and
the oscillatory nucleocytoplasmic shuttling of the transcription factor Msn2 in
yeast. Other examples are the genome-wide periodicity of about 80 min observed
for gene expression in yeast [150, 151] and the ‘‘transcriptional clock’’ based on
the estrogen-receptor-mediated ordered, cyclical recruitment of protein cofactors
involved in target gene transcription [152].
A.
Oscillations of p53 and NF-KB
The tumor suppressor p53 plays an important role in the control of the cell
cycle, and it is inactivated in many types of human tumors. In response to
genomic stress, p53 activation may elicit cell-cycle arrest or apoptotic cell
death, as well as contribute to DNA repair processes. Regulation of p53 is
mediated by its interactions with the protein Mdm2. The binding of Mdm2 to p53
inhibits the transcriptional functions of p53 and also leads to its degradation.
At the same time, p53 stimulates the transcription of the mdm2 gene. These
interactions define a negative feedback loop ensuring that the p53 response
is brought to an end once a p53-activating stress signal has been effectively
dealt with.
The dynamics of the p53-Mdm2 feedback loop was analyzed mathematically
by Lev Bar-Or et al. [153], who pointed out that this negative feedback
regulation can give rise to oscillatory behavior if there is a sufficient delay in the
induction of Mdm2 by p53. They verified experimentally that oscillations of
both p53 and Mdm2 indeed occur on exposure of various cell types to ionizing
radiation. Lahav et al. [154] pursued this study by investigating the dynamics of
p53 and Mdm2 in individual cells. They showed that p53 was expressed in a
series of discrete pulses after DNA damage. The number of p53 pulses, but
not their amplitude, varied in different cells and increased with DNA damage.
Ma et al. [155] recently proposed a model for this ‘‘digital’’ response of
individual cells to DNA damage. The model is based on the coupling of DNA
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albert goldbeter
damage to the p53-Mdm2 oscillator. An alternative model for oscillations in the
p53/Mdm2 module has also been proposed [156].
A negative feedback loop likewise controls the activity of the transcription
factor NF-KB, which is rapidly turned off by a protein inhibitor, I-KB, which
exists under three isoforms denoted a, b, and e. Only the I-KBa isoform
participates in the negative feedback. When NF-KB dissociates from I-KB in
the cytosol, it enters the nucleus, where it induces the transcription of a number
of target genes, one of which codes for the inhibitor I-KBa. Hoffmann et al.
[157] analyzed a computational model based on the interactions of NF-KB and
its inhibitor I-KBa. They predicted and verified experimentally that these
interactions can give rise to oscillatory changes in NF-KB activity characterized
by a period of the order of several hours. Using single-cell time-lapse imaging
and computational modeling, Nelson et al. [158] showed that NF-KB
localization oscillates between the nucleus and the cytosol following cell
stimulation. The frequency of these oscillations decreased with increased I-KBa
transcription. The question of whether the pattern of NF-KB oscillations can
selectively control the expression of certain genes remains open [157–159].
B.
Segmentation Clock
The formation of somites in the course of vertebrate development is associated
with body segmentation. This phenomenon represents a striking example of
spatial pattern in morphogenesis. It has long been suggested that a temporal
periodic process is also at work in somitogenesis, since pairs of somites form
progressively, one at a time, along the presomitic mesoderm (PSM). Thus Cooke
and Zeeman [160] proposed a ‘‘clock and wavefront’’ model, which postulated
the existence of (a) a wavefront moving from the anterior to the posterior end of
the PSM and (b) a clock that would periodically induce the formation of a pair of
somites at the location of the wavefront. Neither the nature of the clock nor the
wavefront-defining process was characterized in this abstract model, which
nevertheless proved highly seminal. Experimental evidence for an oscillator
involved in somitogenesis was later obtained [161, 162]. This oscillator is based
on periodic gene expression and is known as the vertebrate ‘‘segmentation
clock’’ [163–165]. Its period is of the order of 90 min to 2 h, depending on the
organism considered. A unique feature of the segmentation clock is that is links a
temporal oscillation with the formation of a stable spatial pattern, in an important
developmental process [163].
The mechanism of the segmentation clock relies on the negative feedback
exerted on the expression of genes that participate in the signaling pathway
controlled by Notch and other transcription factors [163-165]. Thus, periodic
Notch inhibition by the product of the gene lunatic fringe (Lfng) underlies the
chick segmentation clock [166]. The product of Lfng establishes a negative
feedback loop that results in the periodic inhibition of Notch, which, in turn,
biological rhythms as temporal dissipative structures
277
controls the rhythmic expression of cyclic genes in the chick PSM. This
feedback loop provides a molecular basis for the oscillator underlying the avian
segmentation clock. A model based on such negative autoregulatory feedback
on gene expression (Fig. 4) confirms that it can produce sustained oscillations
[167].
Subsequent experimental studies have shown that another signal pathway
controlled by WNT may drive oscillations in the Notch pathway [168]. The Wnt
pathway is also regulated by negative feedback and could thus give rise to
oscillatory behavior (Fig. 5), as shown by a modeling study [167]. Oscillations
could be transduced from one pathway to the other by a common intermediate
such as the protein kinase GSK3, which is involved both in Wnt and Notch
signaling.
There is thus a multiplicity of negative feedback processes that could, in
principle, give rise to oscillations in Notch signaling, and on which the
Notch
Negative feedback
GSK3
NICD
degradation
Wnt oscillator
NICD-P
nuclear NICD-P
fringe mRNA
hairy mRNA
FRINGE
Figure 4. A negative feedback mechanism in the Notch signaling pathway can give rise to
periodic expression of genes such as Lunatic fringe (Lnfg). This negative feedback on transcription is
thought to play a key role in the segmentation clock controlling somatogenesis in vertebrates (see
Section VIII.B). Upon cleavage, the Notch ligand produces the form NICD, which, after
phosphorylation, possibly by the kinase GSK3, migrates to the nucleus where it induces the
expression of genes like Hairy and Lfng. The FRINGE protein inhibits the cleavage of Notch into
NICD. The model based on this negative feedback regulation shows that it can give rise to sustained
oscillations.
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albert goldbeter
WNT
DSH
-
GSK3 + Axin
B-catenin
destruction
complex
B-catenin-P
degradation
nuclear B-catenin
Axin mRNA
Axin
Negative
feedback
Figure 5. A negative feedback mechanism in the Wnt signaling pathway can also give rise to
periodic gene expression. This negative feedback on transcription could form the core of the
segmentation clock mechanism by driving oscillations in the Notch pathway. Wnt signaling activates
the ligand DSH which inhibits the b-catenin destruction complex formed by the protein Axin and the
kinase GSK3. This destruction complex phosphorylates b-catenin and thereby marks it for
degradation. Negative feedback originates from the fact that b-catenin induces the expression of the
Axin gene; translation of Axin mRNA results in the accumulation of the Axin protein, which leads to
b-catenin degradation. Here again a model based on this negative feedback regulation shows that it
can produce sustained oscillations. Coupling of the Notch oscillator to the Wnt oscillator could be
mediated by oscillations in the activity of the kinase GSK3.
segmentation clock could be based [165]. The possibility of sustained
oscillations resulting from such negative feedback loops has been investigated
in theoretical models [169, 170] that emphasize the role of time delays in the
appearance of sustained oscillations. A role for intercellular signaling through
the Notch-Delta pathway has also been pointed out in the model proposed for
the zebra fish segmentation clock [169].
One intriguing aspect of the models for the segmentation clock is their link
with mechanisms proposed for circadian rhythms. Both types of oscillations are
based on negative autoregulation of gene expression. The question arises as to
how similar mechanisms produce oscillations with a period in the range
30 min to 2 h for the segmentation clock and around 24 h for circadian rhythms.
It would be interesting to further characterize the differences that lead to a
10-fold change in period in the two situations. Such differences may pertain, for
biological rhythms as temporal dissipative structures
279
example, to the half-life of proteins or mRNAs or to the post-transcriptional
regulation of proteins involved in the oscillatory mechanism.
Oscillations of the segmentation clock with a period of 2 h have also been
observed in fibroblast cell cultures following serum shock. There also,
oscillations in the expression of the gene Hes1 related to the Notch pathway
have been attributed to negative feedback on transcription [171]. The periodic
operation of the segmentation clock was recently demonstrated in cells of the
PSM, where intercellular coupling is needed to prevent damping of the
oscillations [172].
Progress has also been made on the experimental characterization of the
biochemical process that mediates the anterior to posterior progression of the
determination wavefront along the PSM during somitogenesis. Fibroblast
growth factor (FGF) signaling is involed in the coupling between the
segmentation clock and the formation of somites [173]. Moreover, a gradient
in fgf mRNA, starting at the posterior end of the PSM, extends in the direction
of the anterior end where its progressive degradation results in the anterior to
posterior movement of the wavefront [174]. The study of a theoretical model
has recently shown [175] that the bistability assumed in the clock and wavefront
model [160] could originate from the antagonistic gradients of mutually
inhibiting FGF and retinoic acid [176] along the PSM. Mutual inhibition is
known to give rise to bistability, as demonstrated, for example, in a synthetic
genetic network [177].
C.
Nucleocytoplasmic Oscillations of the Transcription
Factor Msn2 in Yeast
In the yeast Saccharomyces cerevisiae, two related transcriptional activators
Msn2 and Msn4 are activated in various stress conditions. These proteins are
located in the cytoplasm, but they migrate to the nucleus upon activation.
Translocation to the nucleus is inhibited by high activity of the cAMP-PKA
pathway. Using time-lapse video microscopy on single cells, Jacquet et al. [87]
followed the kinetics of translocation of Msn2 fused to the Green fluorescence
protein (GFP). They showed that light emission of the microscope is sufficient to
induced migration of Msn2 to the nucleus and therefore is sensed as a stress by
the cell. Unexpectedly, the population of Msn2 molecules displayed an
oscillatory behavior, shuttling repetitively between nucleus and cytoplasm
upon light stress, with a periodicity of the order of several minutes. The
phenomenon presents a large variability between individual cells. Upon
additional stress the oscillatory behavior is maintained but the average time
spent in the nucleus is increased. A plausible theoretical model was proposed to
account for such oscillations, based on the hypothesis that this transcriptional
regulator is involved in an autoregulatory loop controlling its nuclear localization
[87].
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albert goldbeter
An alternative possibility is that a biochemical oscillator controls the
periodic shuttling of Msn2 between cytosol and nucleus. So far the existence of
such a putative biochemical oscillator driving Msn2 oscillatory shuttling has not
been substantiated by experimental observations. Among possible biochemical
oscillators with periods in the range of minutes, glycolysis could be a natural
candidate (see Section IV), but glycolytic oscillations are controlled by glucose
rather than stress. Calcium oscillations have not been observed in yeast, and
their occurrence may be precluded by the fact that this organism lacks InsP3
receptors. Recent observations [178] suggest that oscillatory shuttling of Msn2
may well be controlled by oscillations of cAMP, via the protein kinase PKA,
which is activated by cAMP. It appears indeed that the cellular localization of
Msn2 is governed through phosphorylation by PKA.
As shown by a theoretical model [178], the periodic variation of cAMP could
originate from the negative feedback exerted via PKA on the synthesis of cAMP
(Fig. 6). Thus, PKA could exert its negative control by inactivating through
phosphorylation the GAP protein, which is involved in the activation of adenylate cyclase, or by activating the enzyme phosphodiesterase, which degrades
cAMP. Such a mechanism producing intracellular oscillations of cAMP could
operate in other cell types. Thus it is closely related to the intracellular
mechanism proposed for cAMP oscillations in Dictyostelium [84, 85].
Oscillations of cAMP were also proposed to underlie pulsatile hormone release
in GnRH secreting cells [179]. The evidence for cAMP oscillations in yeast so
far remains indirect, since the variations of this metabolite cannot be followed
continuously. The oscillatory nucleocytoplasmic shuttling of Msn2 nevertheless
provides an indirect sign that cAMP might oscillate in yeast.
IX.
COMPLEX OSCILLATORY BEHAVIOR
The transition from simple to complex oscillatory phenomena is often observed
in biochemical and cellular systems. Thus, bursting represents one type of
complex oscillations that is particularly common in neurobiology [29]. An active
phase of spike generation is followed by a quiescent phase, after which a new
active phase begins. Mathematical models throw light on the conditions that
generate such complex periodic oscillations [180]. Chaos is another common
mode of complex oscillatory behavior that has been studied intensively in
physical, chemical, and biological systems [31, 122, 181]. These irregular
oscillations are characterized by their sensitivity to initial conditions, which
accounts for the unpredictable nature of chaotic dynamics.
Yet another type of complex oscillatory behavior involves the coexistence
of multiple attractors. Hard excitation refers to the coexistence of a stable
steady state and a stable limit cycle—a situation that might occur in the case
of circadian rhythm suppression discussed in Section VI. Two stable limit
biological rhythms as temporal dissipative structures
281
GEF i
GEF a
Ras GDP
Ras GTP
GAP a
Cyclase i
Cyclase a
cAMP PDE
GAP i
R2C2
R2 + 2C
Figure 6. Regulation of the cAMP-PKA pathway in yeast. Synthesis of cAMP by adenylate
cyclase is enhanced when this enzyme is activated by Ras-GTP, the active form of the Ras protein.
The transformation of the inactive form Ras-GDP into Ras-GTP is triggered by the active form
GEFa of the GEF protein, while the reverse, inactivating step is elicited by the active form GAPa of
the GAP protein. Protein kinase A is activated when cAMP binds to the holoenzyme form R2C2 and
frees the active, catalytic subunit C. Negative feedback originates from the fact that the catalytic
subunit C of PKA phosphorylates and thereby activates the inactive form GAPi into GAPa. This
leads to a decrease in Ras-GTP and, subsequently, to a drop in cAMP. A model based on this
negative feedback regulation shows the possibility of sustained oscillations in cAMP and PKA
activity [178]. These oscillations could be triggered by the stress-induced inactivation of GEFa.
Stress-induced, nucleocytoplasmic oscillations of the transcription factor Msn2 in yeast would
originate from the control of its subcellular localization through phosphorylation by the catalytic
subunit of PKA (see Section VIII.C).
cycles may also coexist, separated by an unstable limit cycle. This phenomenon, referred to as birhythmicity [182], is the oscillatory counterpart of
bistability in which two stable steady states, separated by an unstable state,
coexist. Birhythmicity was predicted theoretically before being observed
experimentally.
The study of models indicates the existence of two main routes to complex
oscillatory phenomena. The first relies on forcing a system that displays simple
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albert goldbeter
periodic oscillations by a periodic input [122]. In an appropriate range of input
frequency and amplitude, one can often observe the transition from simple to
complex oscillatory behavior such as bursting and chaos. For other frequencies
and amplitudes of the forcing, entrainment or quasiperiodic oscillations occur.
Because circadian rhythms are naturally subjected to periodic forcing by LD
cycles, the possibility arises that such forcing might lead to chaos. Such a
situation would be detrimental to the organism, since entrainment by the LD
cycle is precisely the most conspicuous function of circadian rhythms.
Numerical simulations of a model for the circadian clock indicate that
entrainment, quasiperiodic oscillations, and chaos may indeed occur, depending
on the magnitude of the periodic changes induced by the LD cycle in the lightsensitive parameter. The waveform of the forcing, however, is also important
since the domain of entrainment enlarges at the expense of chaos when the
periodic variation in the light-sensitive parameter changes from square wave to
sinusoidal [183]. Given that the real LD cycle differs from a square wave, the
possibility that chaotic dynamics in the circadian control system seems unlikely
in natural conditions.
Complex oscillations can also occur in autonomous systems that operate in a
constant environment. The study of models for a variety of cellular oscillations
shows that complex oscillatory phenomena may arise through the interplay
between several instability-generating mechanisms, each of which is capable of
producing sustained oscillations [31, 182]. The case of Ca2þ signaling is
particularly revealing because of the multiplicity of feedback mechanisms that
could potentially be involved in the onset of oscillations. Thus, among the many
nonlinear processes that could take part in an instability-generating loop are
(1) Ca2þ -induced Ca2þ release, (2) desensitization of the InsP3 receptor, (3)
bell-shaped dependence of the InsP3 receptor on Ca2þ that reflects its activation
and inhibition at different Ca2þ levels, (4) capacitative Ca2þ entry, (5) PLC or/
and InsP3 3-kinase activation by Ca2þ, (6) control of Ca2þ by mitochondria, (7)
G-protein regulation by Ca2þ, and (8) coupling of the membrane potential to
cytosolic Ca2þ . Several models in which at least two of these regulatory
processes are coupled were shown to admit birhythmicity, bursting, or chaotic
oscillations [62, 184–187]. Chaotic dynamics has been observed experimentally
in glycolysis in autonomous conditions [188], presumably as a result of the
interplay between two endogenous instability-generating mechanisms.
X.
CONCLUDING REMARKS
Since the onset of studies on dissipative structures at the end of the 1960s [12, 14,
15, 18, 189], the field devoted to the investigation of nonequilibrium structures in
chemistry, physics, and the life sciences has grown tremendously. In this chapter
I focused on temporal dissipative structures in biology, a field where they
biological rhythms as temporal dissipative structures
283
abound. My aim was to provide an overview of oscillatory processes in
biological systems, including recent developments and new examples. Rhythmic
phenomena are common at all levels of biological organization because the
thermodynamic and kinetic conditions for the occurrence of rhythmic behavior
are particularly well-satisfied [15, 18]. Indeed, biological systems are open, since
they exchange matter and energy with their surroundings; they often function far
from equilibrium; and their evolution is governed by nonlinear kinetic laws.
The sources of nonlinearity are manifold in biological systems. Besides
cooperativity, a major source of nonlinearity is provided by the variety of
regulatory processes encountered at the cellular level (see Table III). The
existence of regulatory feedback is thus the main reason why rhythmic behavior
is among the most conspicuous properties of living organisms. Positive
feedback is generally associated with multiple steady states [30], while negative
feedback is capable of generating oscillations provided a minimum delay in the
negative feedback loop exists. Inhibition by the product is indeed at the core of
many rhythmic phenomena in biological systems. To name but a few among
those discussed in this chapter, circadian rhythms, oscillations in NF-KB or p53,
and the segmentation clock all appear to be based on direct or indirect, negative
autoregulation. Negative feedback is also involved in the Repressilator, which is
a synthetic oscillatory gene circuit consisting of three repressors coupled in a
cyclical manner [26, 190]. Beyond the particular nature of the molecules
involved, mathematical models help to establish links between various
oscillatory mechanisms based on inhibitory feedback [170].
In oscillatory mechanisms based on negative feedback, a positive effect is
generally needed to sustain periodic behavior, but it can just be an induction
process—as is the case for the action of CLOCK/BMAL1 in the circadian
clock—and does not need to take the form of positive feedback. Positive
feedback is self-amplifying and amounts to activation by the product. Such
mode of regulation is less common than negative feedback but plays a key role
in the origin of several biological rhythms. Besides oscillations of the
membrane potential in nerve and muscle cells, which were not considered in
detail in this chapter devoted to cellular rhythms of nonelectrical nature,
examples of rhythms based on positive feedback include glycolytic oscillations,
Ca2þ spiking, and cAMP oscillations in Dictyostelium cells.
To give rise to oscillatory behavior instead of a biochemical explosion, selfamplification must, however, be coupled to a limiting process. Such a limiting
process can be viewed as a form of negative feedback because it occurs as a
consequence of the positive feedback that precedes it. Thus, in the case of
glycolytic oscillations, the activation of phosphofructokinase by a reaction
product is followed by a counteracting fall in the rate of the enzymatic
reaction, due to the enhanced substrate consumption associated with enzyme
activation. In Ca2þ pulsatile signaling, the explosive rise in cytosolic Ca2þ due
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albert goldbeter
to Ca2þ -induced Ca2þ release from the endoplasmic reticulum is limited by the
emptying of the store and by the inhibition of the Ca2þ channel at high levels of
cytosolic Ca2þ . Likewise, in Dictyostelium, the rapid rise in cAMP due to
positive feedback necessarily leads to its subsequent decrease through cAMPinduced desensitization of the cAMP receptor on the plasma membrane.
Sometimes a mixture of positive and negative feedback can produce
relaxation oscillations based on bistability. This situation is illustrated by the
oscillations in cdk1 activity, which drive the early cell division cycles in
amphibian embryos (see Section VII). While oscillations may originate from the
negative feedback exerted by the cyclin-dependent kinase cdk1 (cdc2) through
activation of the cyclin degradation pathway [133], bistability has been shown
to arise from positive feedback of cdk1 through activation of phosphatase cdc25
and inhibition of kinase wee1, which respectively activate and inhibit cdk1. This
bistability, coupled to the negative feedback loop, results in repetitive cycles of
hysteresis, which correspond to robust, sustained oscillations of cdk1 activity
[135–137].
A striking property that is well illustrated by recently discovered examples of
rhythms in genetic networks is that multiple oscillatory mechanisms can coexist
in a given biological system. This is exemplified by the case of circadian
rhythms discussed in Section VI and by the vertebrate segmentation clock
considered in Section VIII. In the latter system, several negative feedback loops
are present in the Notch or Wnt signaling pathways, providing for multiple
potential sources of oscillatory behavior [165]. The reasons for such a
multiplicity of oscillatory mechanisms may be manifold. One may be to provide
redundancy so that back-up mechanisms are available in case of failure of one
of the regulatory loops. That a feedback loop exists does not guarantee, however, that it operates in the parameter domain producing sustained oscillations.
Even when multiple feedback loops are present within a system, some of them
may mot be capable of producing oscillations by themselves because the values
of the parameters that characterize the feedback module correspond to the
evolution toward a stable steady state. Another reason for the multiplicity of
instability-generating mechanisms may simply be to strengthen the oscillations
by enlarging the domain of rhythmic behavior in parameter space.
As recalled in Section IX, models indicate that the interplay between several
endogenous mechanisms may give rise to complex oscillations in the form of
bursting or chaos. The multiplicity of oscillatory mechanisms within a given
metabolic or genetic network may therefore bear, positively—as in the case of
bursting oscillations or, adversely, as perhaps in the case of chaos—on the
physiological functions of cellular rhythms. Even in the presence of interactions
between multiple instability-generating mechanisms, however, simple periodic
behavior or oscillations of the bursting type remain more common than chaos in
parameter space.
biological rhythms as temporal dissipative structures
285
In several instances, intracellular oscillations are linked to the formation of
spatial or spatiotemporal patterns, as shown by the role of the vertebrate
segmentation clock in somitogenesis, by the propagation of intra- or
intercellular waves in Ca2þ signaling in many cell types, and by cAMP
signaling in Dictyostelium amoebae. Recent observations on the occurrence of
intracellular waves in activated leukocytes [191] provide yet another example of
close link between temporal and spatial organization at the cellular level. The
close intertwining of spatial and temporal patterns led Duboule [192] to write
that ‘‘animal development is, in fact, nothing but time.’’
Since the initial impetus given by Ilya Prigogine to the study of oscillatory
phenomena in chemical and biological systems, the number of examples of
periodic behavior has grown immensely. While some examples were known for
long, new ones were added to the list, which will undoubtedly continue to
expand at an increasing pace. The views of Ilya Prigogine on nonequilibrium
self-organization in the form of dissipative structures provide a conceptual
framework that allows us to unify the multifarious rhythms that occur in
biological systems with periods spanning more than 10 orders of magnitude.
This global perspective underlines the links between rhythmic phenomena
occurring in widely different biological settings, from genetic to metabolic and
neural networks and from cell to animal populations.
Acknowledgments
I wish to dedicate this chapter to the memory of Ilya Prigogine, who triggered my interest in the
modeling of rhythmic phenomena in biological systems and deeply influenced this research by the
originality of his views and his unwavering enthusiasm and generosity. To this homage I wish to
associate the name of Benno Hess, who did much to extend the study of nonequilibrium selforganization phenomena to the field of biological sciences. Thanks are due to members of my
research group for fruitful discussions. This work was supported by grant #3.4636.04 from the Fonds
de la Recherche Scientifique Médicale (F.R.S.M., Belgium) and by the European Union through the
Network of Excellence BioSim, Contract No. LSHB-CT-2004-005137. The chapter was prepared
while the author held a Chaire Internationale de Recherche Blaise Pascal de l’Etat et de la Région Ilede-France, gérée par la Fondation de l’Ecole Normale Supérieure in the Institute of Genetics and
Microbiology at the University of Paris-Sud 11 (Orsay, France).
References
1. W. C. Bray, A periodic chemical reaction and its mechanism. J. Am. Chem. Soc. 43, 1262
(1921).
2. B. van der Pol and J. van der Markt, The heart beat considered as a relaxation oscillation, and an
electrical model of the heart. Philos. Mag. 6, 763–775 (1928).
3. G. Nicolis and J. Portnow, Chemical oscillations. Chem. Rev. 73, 365–384 (1973).
4. R. M. Noyes and R. J. Field, Oscillatory chemical reactions. Annu. Rev. Phys. Chem. 25, 95–119
(1974).
286
albert goldbeter
5. A. J. Lotka, Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc. 42,
1595 (1920).
6. V. Volterra, Fluctuations in the abundance of a species considered mathematically. Nature 118,
558–560 (1926).
7. A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane currents and
its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117, 500–544
(1952).
8. A. H. Huxley, lon movements during nerve activity. Ann. N.Y. Acad. Sci. 81, 221–246 (1959).
9. A. M. Turing, The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72
(1952).
10. I. Prigogine and R. Balescu, Phénomènes cycliques dans la thermodynamique des processus
irréversibles. Bull. Cl. Sci. Acad. R. Belg. XLII, 256–265 (1956).
11. J. J. Tyson, Some further studies of nonlinear oscillations in chemical systems. J. Chem. Phys.
58, 3919–3930 (1973).
12. R. Lefever, G. Nicolis, and I. Prigogine, On the occurrence of oscillations around the steady
state in systems of chemical reactions far from equilibrium. J. Chem. Phys. 47, 1045–1047
(1967).
13. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, John Wiley & Sons,
New York, 1967.
14. I. Prigogine, Structure, dissipation and life, in Theoretical Physics and Biology. M. Marois, ed.,
North-Holland, Amsterdam, pp. 23–52, 1969.
15. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems. From Dissipative
Structures to Order through Fluctuations, John Wiley & Sons, New York, 1977.
16. R. Lefever, G. Nicolis, and P. Borckmans, The Brusselator: It does oscillate all the same.
J. Chem. Soc. Faraday Trans. 1 84, 1013–1023 (1988).
17. N. Minorsky, Nonlinear Oscillations, Van Nostrand, Princeton, NJ, 1962.
18. P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations,
John Wiley & Sons, New York.
19. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations
of Vector Fields, Springer, New York, 1983.
20. R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membranes,
Biophys. J. 1, 445–466 (1961).
21. R. M. May, Limit cycles in predator–prey communities. Science 177, 900–902 (1972).
22. A. M. Zhabotinsky, Periodic process of the oxidation of malonic acid in solution. Study of the
kinetics of Belousov’s reaction. Biofizika 9, 1306 (1964).
23. R. J. Field, E. Köros, and R. M. Noyes, Oscillations in chemical systems. II. Thorough analysis
of temporal oscillation in the bromate–cerium–malonic acid system. J. Am. Chem. Soc. 94,
8649–8664 (1972).
24. S. Nakamura, K. Yokota, and I. Yamazaki, Sustained oscillations in a lactoperoxidase, NADPH
and O2 system. Nature 222 794 (1969).
25. A. Fessard, Propriétés Rythmiques de la Matière Vivante, Hermann, Paris, 1936.
26. M. B. Elowitz and S. Leibler, A synthetic oscillatory network of transcriptional regulators.
Nature 403, 335–338 (2000).
27. R. Llinas, The intrinsic electrophysiological properties of mammalian neurons: a new insight
into CNS function, Science 242, 1654–1664 (1988).
biological rhythms as temporal dissipative structures
287
28. D. DiFrancesco, Pacemaker mechanisms in cardiac tissue. Annu. Rev. Physiol. 55, 455–472
(1993).
29. A. Destexhe and T. J. Sejnowski, Interactions between membrane conductances underlying
thalamocortical slow wave oscillations. Physiol. Rev 83, 1401–1453 (2003).
30. R. Thomas and R. d’Ari, Biological Feedback, CRC Press, Boca Raton, FL, 1990.
31. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms. The Molecular Bases of
Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, UK, 1996.
32. A. Goldbeter, Computational approaches to cellular rhythms. Nature 420, 238–245 (2002).
33. C. Koch and I. Segev, eds., Methods in Neuronal Modeling. From Synapses to Networks, 2nd
ed., MIT Press, Cambridge, MA, 1998.
34. J. P. Keener and J. Sneyd, Mathematical Physiology, Springer, New York, 1998.
35. D. Noble, Modeling the heart—from genes to cells to the whole organ. Science 295, 1678–1682
(2002).
36. B. Chance, B. Schoener, and S. Elsaesser, Control of the waveform of oscillations of the
reduced pyridine nucleotide level in a cell-free extract. Proc. Natl. Acad. Sci. USA 52, 337–341
(1964).
37. B. Hess, K. Brand, and K. Pye, Continuous oscillations in a cell-free extract of S. carlsbergensis. Biochem. Biophys. Res. Commun. 23, 102–108 (1966).
38. M. F. Madsen, S. Dano, and P. G. Sorensen, On the mechanisms of glycolytic oscillations in
yeast. FEBS J. 272, 2648–2660 (2005).
39. H. F. Chou, N. Berman, and E. Ipp, Oscillations of lactate released from islets of Langerhans:
Evidence for oscillatory glycolysis in b-cells. Am. J. Physiol. 262, E800–E805 (1992).
40. B. Hess and A. Boiteux, Oscillatory phenomena in biochemistry. Annu. Rev. Biochem. 40, 237–
258 (1971).
41. A. Goldbeter and S. R. Caplan, Oscillatory enzymes. Annu. Rev. Biophys. Bioeng. 5, 449–476
(1976).
42. K. A. Reijenga, H. V. Westerhoff, B. N. Kholodenko, and J. L. Snoep, Control analysis for
autonomously oscillating biochemical networks. Biophys. J. 82, 99–108 (2002).
43. J. Higgins, A chemical mechanism for oscillation of glycolytic intermediates in yeast cells.
Proc. Natl. Acad. Sci. USA 51, 989–994 (1964).
44. E. E. Sel’kov, Self-oscillations in glycolysis. 1. A simple kinetic model. Eur. J. Biochem. 4, 79–
86 (1968).
45. A. Goldbeter and R. Lefever, Dissipative structures for an allosteric model. Application to
glycolytic oscillations. Biophys. J. 12, 1302–1315 (1972).
46. A. Boiteux, A. Goldbeter, and B. Hess, Control of oscillating glycolysis of yeast by stochastic,
periodic, and steady source of substrate: a model and experimental study. Proc. Natl. Acad. Sci.
USA 72, 3829–3833 (1975).
47. S. Dano, P. G. Sorensen, and F. Hynne, Sustained oscillations in living cells. Nature 402, 320–
322 (1999).
48. D. Garfinkel and B. Hess, Metabolic control mechanisms. VII. A detailed computer model of
the glycolytic pathway in ascites cells. J. Biol. Chem. 239, 971–983 (1964).
49. Y. Termonia and J. Ross, Oscillations and control features in glycolysis: Numerical analysis of a
comprehensive model. Proc. Natl. Acad. Sci. USA 78, 2952–2956 (1981).
50. F. Hynne, S. Dano, and P. G. Sorensen, Full-scale model of glycolysis in Saccharomyces
cerevisiae. Biophys. Chem. 94, 121–163 (2001).
288
albert goldbeter
51. J. Wolf, J. Passarge, O. J. Somsen, J. L. Snoep, R. Heinrich, and H. V. Westerhoff, Transduction
of intracellular and intercellular dynamics in yeast glycolytic oscillations. Biophys. J. 78, 1145–
1153 (2000).
52. P. Richard, B. M. Bakker, B. Teusink, K. Van Dam, H. V. Westerhoff, Acetaldehyde mediates
the synchronization of sustained glycolytic oscillations in populations of yeast cells. Eur.
J. Biochem. 235, 238–241 (1996).
53. K. Tornheim, Are metabolic oscillations responsible for normal oscillatory insulin secretion?
Diabetes 46, 1375–1380 (1997).
54. M. G. Pedersen, R. Bertram, and A. Sherman, Intra- and inter-islet synchronization of
metabolically driven insulin secretion. Biophys. J. 89, 107–119 (2005).
55. K. Wierschem and R. Bertram, Complex bursting in pancreatic islets: a potential glycolytic
mechanism. J. Theor. Biol. 228, 513–521 (2004).
56. M. J. Berridge, Elementary and global aspects of calcium signalling. J. Physiol. (London) 499,
291–306 (1997).
57. A. Fabiato, Calcium-induced release of calcium from the cardiac sarcoplasmic reticulum. Am.
J. Physiol. 245, C1–C14 (1983).
58. M. Wakui, Y. V. Osipchuk, and O. H. Petersen, Receptor-activated cytoplasmic Ca2þ spiking
mediated by inositol trisphosphate isdue toCa2þ -induced Ca2þ release.Cell63, 1025–1032 (1990).
59. T. Meyer and L. Stryer, Molecular model for receptor-stimulated calcium spiking. Proc. Natl.
Acad. Sci. USA 85, 5051–5055 (1988).
60. A. Goldbeter, G. Dupont, and M. J. Berridge, Minimal model for signal-induced Ca2þ
oscillations and for their frequency encoding through protein phosphorylation. Proc. Natl.
Acad. Sci. USA 87, 1461–1465 (1990).
61. G. W. De Young and J. Keizer, A single-pool inositol 1,4,5-trisphosphate-receptor-based model
for agonist-stimulated oscillations in Ca2þ concentration. Proc. Natl. Acad. Sci. USA 89, 9895–
9899 (1992).
62. S. Schuster, M. Marhl, and T. Höfer, Modelling of simple and complex calcium oscillations.
From single-cell responses to intercellular signalling. Eur. J. Biochem. 269, 1333–1355 (2002).
63. G. Dupont and A. Goldbeter, Properties of intracellular Ca2þ waves generated by a model based
on Ca2þ -induced Ca2þ release. Biophys. J. 67, 2191–2204 (1994).
64. J. Sneyd, A. C. Charles, and M. J. Sanderson, A model for the propagation of intercellular
calcium waves. Am. J. Physiol. 266, C293–C302 (1994).
65. G. Dupont, T. Tordjmann, C. Clair, S. Swillens, M. Claret, and L. Combettes, Mechanism of
receptor-oriented intercellular calcium wave propagation in hepatocytes. FASEB J. 14, 279–289
(2000).
66. S. Swillens, G. Dupont, L. Combettes, and P. Champeil, From calcium blips to calcium puffs:
Theoretical analysis of the requirements for interchannel communication. Proc. Natl. Acad. Sci.
USA 96, 13750–13755 (1999).
67. N. C. Spitzer, N. J. Lautermilch, R. D. Smith, and T. M. Gomez, Coding of neuronal
differentiation by calcium transients. BioEssays 22, 811–817 (2000).
68. P. De Koninck and H. Schulman, Sensitivity of CaM kinase II to the frequency of Ca2þ
oscillations. Science 279, 227–230 (1998).
69. G. Dupont, G. Houart, and P. De Koninck, Sensitivity of CaM kinase II to the frequency of Ca2þ
oscillations: A simple model. Cell Calcium 34, 485–497 (2003).
70. Y. V. Gorbunova and N. C. Spitzer, Dynamic interactions of cyclic AMP transients and
spontaneous Ca2þ spikes. Nature 418, 93–96 (2002).
biological rhythms as temporal dissipative structures
289
71. G. Dupont and L. Combettes, Modelling the effect of specific inositol 1,4,5-trisphosphate
receptor isoforms on cellular Ca2þ signals. Biol. Cell 98, 171–182 (2006).
72. J. Sneyd, K. Tsaneva-Atanasova, V. Reznikov, Y. Bai, M. J. Sanderson, and D. I. Yule, A method
for determining the dependence of calcium oscillations on inositol trisphosphate oscillations.
Proc. Natl. Acad. Sci. USA 103, 1675–1680 (2006).
73. M. Beltramello, V. Piazza, F. F. Bukauskas, T. Pozzan, and F. Mammano, Impaired permeability
to Ins(1,4,5)P3 in a mutant connexin underlies recessive hereditary deafness. Nat. Cell. Biol. 7,
63–69 (2005).
74. P. Uhlen, P. M. Burch, C. I. Zito, M. Estrada, B. E. Ehrlich, and A. M. Bennett, Gain-offunction/Noonan syndrome SHP-2/Ptpn11 mutants enhance calcium oscillations and impair
NFAT signaling. Proc. Natl. Acad. Sci. USA 103, 2160–2165 (2006).
75. D. Dormann, J. Y. Kim, P. Devreotes, and C. J. Weijer, cAMP receptor affinity controls
wave dynamics, geometry and morphogenesis in Dictyostelium. J. Cell. Sci. 114, 2513–2523
(2001).
76. J. L. Martiel and A. Goldbeter, A model based on receptor desensitization for cyclic AMP
signaling in Dictyostelium cells. Biophys. J. 52, 807–828 (1987).
77. Y. Tang and H. G. Othmer, Excitation, oscillations and wave propagation in a G-protein-based
model of signal transduction in Dictyostelium discoideum. Philos. Trans. R. Soc. Lond. B. Biol.
Sci. 349, 179–195 (1995).
78. Y. X. Li and A. Goldbeter, Frequency encoding of pulsatile signals of cyclic AMP based on
receptor desensitization in Dictyostelium cells. J. Theor. Biol. 146, 355–367 (1990).
79. A. Goldbeter and L. A. Segel, Control of developmental transitions in the cyclic AMP signaling
system of Dictyostelium discoideum. Differentiation 17, 127–135 (1980).
80. H. Levine, I. Aranson, L. Tsimring, and T. V. Truong, Positive genetic feedback governs cAMP
spiral wave formation in Dictyostelium. Proc. Natl. Acad. Sci. USA 93, 6382–6386 (1996).
81. E. Palsson and E. C. Cox, Origin and evolution of circular waves and spirals in Dictyostelium
discoideum territories. Proc. Natl. Acad. Sci. USA 93, 1151–1155 (1996).
82. J. Lauzeral, J. Halloy, and A. Goldbeter, Desynchronization of cells on the developmental path
triggers the formation of spiral waves of cAMP during Dictyostelium aggregation. Proc. Natl.
Acad. Sci. USA 94, 9153–9158 (1997).
83. T. Bretschneider, F. Siegert, and C. J. Weijer, Three-dimensional scroll waves of cAMP could
direct cell movement and gene expression in Dictyostelium slugs. Proc. Natl. Acad. Sci. USA
92, 4387–4391 (1995).
84. M. T. Laub and W. F. Loomis, A molecular network that produces spontaneous oscillations in
excitable cells of Dictyostelium. Mol. Biol. Cell 9, 3521–3532 (1998).
85. M. Maeda, S. Lu, G. Shaulsky, Y. Miyazaki, H. Kuwayama, Y. Tanaka, A. Kuspa, and W. F.
Loomis, Periodic signaling controlled by an oscillatory circuit that includes protein kinases
ERK2 and PKA. Science 304, 875–878 (2004).
86. S. Sawai, P. A. Thomason, and E. C. Cox, An autoregulatory circuit for long-range
self-organization in Dictyostelium cell populations. Nature 433, 323–326 (2005).
87. M. Jacquet, G. Renault, S. Lallet, J. De Mey, A. Goldbeter, Oscillatory nucleocytoplasmic
shuttling of the general stress response transcriptional activators Msn2 and Msn4 in
Saccharomyces cerevisiae. J. Cell Biol. 161, 497–505 (2003).
88. D. J. Chadwick and J. A. Goode, eds., Mechanisms and Biological Significance of Pulsatile
Hormone Secretion. Novartis Foundation Symposium 227, John Wiley & Sons, Chichester, UK,
2000.
290
albert goldbeter
89. E. Knobil, Patterns of hormone signals and hormone action. N. Engl. J. Med. 305, 1582–1583
(1981).
90. P. E. Belchetz, T. M. Plant, Y. Nakai, E. J. Keogh, and E. Knobil, Hypophysial responses to
continuous and intermittent delivery of hypothalamic gonadotropin-releasing hormone.
Science 202, 631–633 (1978).
91. P. C. Hindmarsh, R. Stanhope, M. A. Preece, and C. G. D. Brook, Frequency of administration
of growth hormone—An important factor in determining growth response to exogenous growth
hormone. Horm. Res. 33(Suppl. 4), 83–89 (1990).
92. Y. X. Li and A. Goldbeter, Frequency specificity in intercellular communication: The influence
of patterns of periodic signalling on target cell responsiveness. Biophys. J. 55, 125–145 (1989).
93. C. Wagner, S. R. Caplan, and G. S. Tannenbaum, Genesis of the ultradian rhythm of GH
secretion: A new model unifying experimental observations in rats. Am. J. Physiol. 275, E1046–
1054 (1998).
94. L. W. Maki and J. Keizer, Mathematical analysis of a proposed mechanism for oscillatory
insulin secretion in perifused HIT-15 cells. Bull. Math. Biol. 57, 569–591 (1995).
95. J. C. Dunlap, Molecular bases for circadian clocks. Cell 96, 271–290 (1999).
96. M. W. Young and S. A. Kay, Time zones: A comparative genetics of circadian clocks. Nat. Rev.
Genet. 2, 702–715 (2001).
97. S. M. Reppert and D. R. Weaver, Coordination of circadian timing in mammals. Nature 418,
935–941 (2002).
98. P. E. Hardin, The circadian timekeeping system of Drosophila. Curr. Biol. 15, R714–R722
(2005).
99. D. Bell-Pedersen, V. M. Cassone, D. J. Earnest, S. S. Golden, P. E. Hardin, T. L. Thomas, and
M. J. Zoran, Circadian rhythms from multiple oscillators: Lessons from diverse organisms. Nat.
Rev. Genet. 6, 544–556 (2005).
100. P. E. Hardin, J. C. Hall, and M. Rosbash, Feedback of the Drosophila period gene product on
circadian cycling of its messenger RNA levels. Nature 343, 536–540 (1990).
101. M. Nakajima, K. Imai, H. Ito, T. Nishiwaki, Y. Murayama, H. Iwasaki, T. Oyama, and T. Kondo,
Reconstitution of circadian oscillation of cyanobacterial KaiC phosphorylation in vitro. Science
308, 414–415 (2005).
102. R. E. Kronauer, D. B. Forger, and M. E. Jewett, Quantifying human circadian pacemaker
response to brief, extended, and repeated light stimuli over the phototopic range. J. Biol.
Rhythms 14, 500–515 (1999).
103. Y., Ouyang, C. R. Andersson, T. Kondo, S. S. Golden, and C. H. Johnson. Resonating circadian
clocks enhance fitness in cyanobacteria. Proc. Natl. Acad. Sci. USA 95, 8660–8664
(1998).
104. D. Gonze, M. Roussel, and A. Goldbeter, A model for the enhancement of fitness in
cyanobacteria based on resonance of a circadian oscillator with the external light–dark cycle.
J. Theor. Biol. 214, 577–597 (2002).
105. B. C. Goodwin, Oscillatory behavior in enzymatic control processes. Adv. Enzyme Regul. 3,
425–438 (1965).
106. P. Ruoff, M. Vinsjevik, C. Monnerjahn, and L. Rensing, The Goodwin model: Simulating the
effect of light pulses on the circadian sporulation rhythm of Neurospora crassa. J. Theor. Biol.
209, 29–42 (2001).
107. A. Goldbeter, A model for circadian oscillations in the Drosophila period protein (PER). Proc.
R. Soc. Lond. B Biol. Sci. 261, 319–324 (1995).
biological rhythms as temporal dissipative structures
291
108. J. C. Leloup and A. Goldbeter, A model for circadian rhythms in Drosophila incorporating
the formation of a complex between the PER and TIM proteins. J. Biol. Rhythms 13, 70–87
(1998).
109. U. Schibler and P. Sassone-Corsi, A web of circadian pacemakers. Cell 111, 919–922 (2002).
110. N. R. Glossop, L. C. Lyons, and P. E. Hardin, Interlocked feedback loops within the Drosophila
circadian oscillator. Science 286, 766–768 (1999).
111. L. P. Shearman, S. Sriram, D. R. Weaver, E. S. Maywood, I. Chaves, B. Zheng, K. Kume, C. C.
Lee, G. T. van der Horst, M. H. Hastings, and S. M. Reppert, Interacting molecular loops in the
mammalian circadian clock. Science 288, 1013–1019 (2000).
112. H. R. Ueda, M. Hagiwara, and H. Kitano, Robust oscillations within the interlocked feedback
model of Drosophila circadian rhythm. J. Theor. Biol. 210, 401–406 (2001).
113. P. Smolen, D. A. Baxter, and J. H. Byrne, Modeling circadian oscillations with interlocking
positive and negative feedback loops. J. Neurosci. 21, 6644–6656 (2001).
114. J. C. Leloup and A. Goldbeter, Toward a detailed computational model for the mammalian
circadian clock. Proc. Natl. Acad. Sci. USA 100, 7051–7056 (2003).
115. D. B. Forger and C. S. Peskin, A detailed predictive model of the mammalian circadian clock.
Proc. Natl. Acad. Sci. USA 100, 14806–14811 (2003).
116. M. Doi, J. Hirayama, and P. Sassone-Corsi, Circadian regulator CLOCK is a histone
acetyltransferase. Cell 125, 497–508 (2006).
117. G. S. Richardson and H. V. Malin, Circadian rhythm sleep disorders: Pathophysiology and
treatment. J. Clin. Neurophysiol. 13, 17–31 (1996).
118. K. L. Toh, C. R. Jones, Y. He, E. J. Eide, W. A. Hinz, D. M. Virshup, L. J. Ptacek, and Y. H. Fu,
An hPer2 phosphorylation site mutation in familial advanced sleep phase syndrome. Science
291, 1040–1043 (2001).
119. Y. Xu, Q. S. Padiath, R. E. Shapiro, C. R. Jones, S. C. Wu, N. Saigoh, K. Saigoh, L. J. Ptacek,
and Y. H. Fu, Functional consequences of a CKIdelta mutation causing familial advanced sleep
phase syndrome. Nature 434, 640–644 (2005).
120. T. Ebisawa, M. Uchiyama, N. Kajimura, K. Mishima, Y. Kamei, M. Katoh, T. Watanabe,
M. Sekimoto, K. Shibui, K. Kim, Y. Kudo, Y. Ozeki, M. Sugishita, R. Toyoshima, Y. Inoue,
N. Yamada, T. Nagase, N. Ozaki, O. Ohara, N. Ishida, M. Okawa, K. Takahashi, and
T. Yamauchi, Association of structural polymorphisms in the human period3 gene with delayed
sleep phase syndrome. EMBO Rep. 2, 342–346 (2001).
121. C. R. Jones, S. S. Campbell, S. E. Zone, F. Cooper, A. DeSano, P. J. Murphy, B. Jones,
L. Czajkowski, and L. J. Ptacek, Familial advanced sleep-phase syndrome: A short-period
circadian rhythm variant in humans. Nat. Med. 5, 1062–1065 (1999).
122. L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University
Press, Princeton, NJ, 1988.
123. M. C. Mackey and J. G. Milton, Dynamical diseases. Ann NY Acad. Sci. 504, 16–32 (1987).
124. M. Nagano, A. Adachi, K. Nakahama, T. Nakamura, M. Tamada, E. Meyer-Bernstein,
A. Sehgal, and Y. Shigeyoshi, An abrupt shift in the day/night cycle causes desynchrony in
the mammalian circadian center. J. Neurosci. 23, 6141–6151 (2003).
125. A. T. Winfree, The Geometry of Biological Time, 2nd ed., Springer, New York, 2001.
126. J. C. Leloup and A. Goldbeter, A molecular explanation for the long-term suppression of
circadian rhythms by a single light pulse. Am. J. Physiol. Regul. Integr. Comp. Physiol. 280,
R1206–R1212 (2001).
127. N. Barkai and S. Leibler, Circadian clocks limited by noise. Nature 403, 267–268 (2000).
292
albert goldbeter
128. D. Gonze, J. Halloy, and A. Goldbeter, Robustness of circadian rhythms with respect to
molecular noise. Proc. Natl. Acad. Sci. USA 99, 673–678 (2002).
129. D. B. Forger and C. S. Peskin, Stochastic simulation of the mammalian circadian clock. Proc.
Natl. Acad. Sci. USA 102, 321–324 (2005).
130. J. M. Vilar, H. Y. Kueh, N. Barkai, and S. Leibler, Mechanisms of noise-resistance in genetic
oscillators. Proc. Natl. Acad. Sci. USA 99, 5988–5992 (2002).
131. A. W. Murray and T. Hunt, The Cell Cycle: An Introduction, Oxford University Press, Oxford,
1993.
132. A. W. Murray and M. W. Kirschner, Cyclin synthesis drives the early embryonic cell cycle.
Nature 339, 275–280 (1989).
133. A. Goldbeter, A minimal cascade model for the mitotic oscillator involving cyclin and cdc2
kinase. Proc. Natl. Acad. Sci. USA 88, 9107–9111 (1991).
134. A. Goldbeter, Modeling the mitotic oscillator driving the cell division cycle. Comments Theor.
Biol. 3, 75–107 (1993).
135. J. R. Pomerening, E. D. Sontag, and J. E. Ferrell, Jr. Building a cell cycle oscillator: Hysteresis
and bistability in the activation of Cdc2. Nat. Cell Biol. 5, 346–351 (2003).
136. W. Sha, J. Moore, K. Chen, A. D. Lassaletta, C. S. Yi, J. J. Tyson, and J. C. Sible, Hysteresis
drives cell-cycle transitions in Xenopus laevis egg extracts. Proc. Natl. Acad. Sci. USA 100,
975–980 (2003).
137. J. R. Pomerening, S. Y. Kim, and J. E. Ferrell, Jr. Systems-level dissection of the cell-cycle
oscillator: Bypassing positive feedback produces damped oscillations. Cell 122, 565–578
(2005).
138. B. Novak and J. J. Tyson, Numerical analysis of a comprehensive model of M-phase control in
Xenopus oocyte extracts and intact embryos. J. Cell Sci. 106, 1153–1168 (1993).
139. B. Novak and J. J. Tyson, A model for restriction point control of the mammalian cell cycle.
J. Theor. Biol. 230, 563–579 (2004).
140. J. J. Tyson and B. Novak, Regulation of the eukaryotic cell cycle: Molecular antagonism,
hysteresis, and irreversible transitions. J. Theor. Biol. 210, 249–263 (2001).
141. J. J. Tyson, K. Chen, and B. Novak, Network dynamics and cell physiology. Nat. Rev. Mol. Cell
Biol. 2, 908–916 (2001).
142. K. C. Chen, L. Calzone, A. Csikasz-Nagy, F. R. Cross, B. Novak, and J. J. Tyson, Integrative analysis of cell cycle control in budding yeast. Mol. Biol. Cell 15, 3841–3862
(2004).
143. M. Swat, A. Kel, and H. Herzel, Bifurcation analysis of the regulatory modules of the
mammalian G1/S transition. Bioinformatics 20, 1506–1511 (2004).
144. B. Novak, Z. Pataki, A. Ciliberto, and J. J. Tyson, Mathematical model of the cell division cycle
of fission yeast. Chaos 11, 277–286 (2001).
145. C. Gérard and A. Goldbeter, In preparation.
146. A. W. Murray and M. W. Kirschner, Dominoes and clocks: The union of two views of the cell
cycle. Science 246, 614–621 (1989).
147. J. Halloy, B. A. Bernard, G. Loussouarn, and A. Goldbeter, Modeling the dynamics of human
hair cycles by a follicular automaton. Proc. Natl. Acad. Sci. USA 97, 8328–8333 (2000).
148. A. Altinok and A. Goldbeter, In preparation.
149. T. Matsuo, S. Yamaguchi, S. Mitsui, A. Emi, F. Shimoda, and H. Okamura, Control mechanism
of the circadian clock for timing of cell division in vivo. Science 302, 255–259 (2003).
biological rhythms as temporal dissipative structures
293
150. R. R. Klevecz, J. Bolen, G. Forrest, and D. B. Murray, A genomewide oscillation in
transcription gates DNA replication and cell cycle. Proc. Natl. Acad. Sci. USA 101, 1200–
1205 (2004).
151. M. W. Young, An ultradian clock shapes genome expression in yeast. Proc. Natl. Acad. Sci.
USA 101, 1118–1119 (2004).
152. R. Metivier, G. Penot, M. R. Hubner, G. Reid, H. Brand, M. Kos, and F. Gannon, Estrogen
receptor-alpha directs ordered, cyclical, and combinatorial recruitment of cofactors on a natural
target promoter. Cell 115, 751–763 (2003).
153. R. Lev Bar-Or, R. Maya, L. A. Segel, U. Alon, A. J. Levine, and M. Oren, Generation of
oscillations by the p53-Mdm2 feedback loop: A theoretical and experimental study. Proc. Natl.
Acad. Sci. USA 97, 11250–11255 (2000).
154. G. Lahav, N. Rosenfeld, A. Sigal, N. Geva-Zatorsky, A. J. Levine, M. B. Elowitz, and U. Alon,
Dynamics of the p53-Mdm2 feedback loop in individual cells. Nat. Genet. 36, 147–150 (2004).
155. L. Ma, J. Wagner, J. J. Rice, W. Hu, A. J. Levine, and G. A. Stolovitzky, A plausible model for
the digital response of p53 to DNA damage. Proc. Natl. Acad. Sci. USA 102, 14266–14271
(2005).
156. A. Ciliberto, B. Novak, and J. J. Tyson, Steady states and oscillations in the p53/Mdm2 network.
Cell Cycle 4, 488–493 (2005).
157. A. Hoffmann, A. Levchenko, M. L. Scott, and D. Baltimore, The IkappaB-NF-kappaB
signaling module: Temporal control and selective gene activation. Science 298, 1241–1245
(2002).
158. D. E. Nelson, A. E. Ihekwaba, M. Elliott, J. R. Johnson, C. A. Gibney, B. E. Foreman,
G. Nelson, V. See, C. A. Horton, D. G. Spiller, S. W. Edwards, H. P. McDowell, J. F. Unitt,
E. Sullivan, R. Grimley, N. Benson, D. Broomhead, D. B. Kell, and M. R. White,
Oscillations in NF-kappaB signaling control the dynamics of gene expression. Science
306, 704–708 (2004).
159. D. Barken, C. J. Wang, J. Kearns, R. Cheong, A. Hoffmann, and A. Levchenko, Comment on
‘‘Oscillations in NF-kappaB signaling control the dynamics of gene expression.’’ Science 308,
52 (2005).
160. J. Cooke and E. C. Zeeman, A clock and wavefront model for control of the number of repeated
structures during animal morphogenesis. J. Theor. Biol. 58, 455–476 (1976).
161. I. Palmeirim, D. Henrique, D. Ish-Horowicz, and O. Pourquié, Avian hairy gene expression
identifies a molecular clock linked to vertebrate segmentation and somitogenesis. Cell 91, 639–
648 (1997).
162. M. Maroto and O. Pourquié, A molecular clock involved in somite segmentation. Curr. Top.
Dev. Biol. 51, 221–248 (2001).
163. O. Pourquié, The segmentation clock: Converting embryonic time into spatial pattern. Science
301, 328–330 (2003).
164. Y. Bessho and R. Kageyama, Oscillations, clocks and segmentation. Curr. Opin. Genet. Dev. 13,
379–384 (2003).
165. F. Giudicelli and J. Lewis, The vertebrate segmentation clock. Curr. Opin. Genet. Dev. 14, 407–
414 (2004).
166. J. K. Dale, M. Maroto, M. L. Dequeant, P. Malapert, M. McGrew, and O. Pourquie, Periodic
notch inhibition by lunatic fringe underlies the chick segmentation clock. Nature 421, 275–278
(2003).
167. A. Goldbeter and O. Pourquié, Unpublished results.
294
albert goldbeter
168. A. Aulehla, C. Wehrle, B. Brand-Saberi, R. Kemler, A. Gossler, B. Kanzler, and B. G.
Herrmann, Wnt3a plays a major role in the segmentation clock controlling somitogenesis. Dev.
Cell. 4, 395–406 (2003).
169. J. Lewis, Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish
somitogenesis oscillator. Curr. Biol. 13, 1398–1408 (2003).
170. N. A. Monk, Oscillatory expression of Hes1, p53, and NF-kappaB driven by transcriptional
time delays. Curr. Biol. 13, 1409–1413 (2003).
171. H. Hirata, S. Yoshiura, T. Ohtsuka, Y. Bessho, T. Harada, K. Yoshikawa, and R. Kageyama,
Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop. Science
298, 840–843 (2002).
172. Y. Masamizu, T. Ohtsuka, Y. Takashima, H. Nagahara, Y. Takenaka, K. Yoshikawa,
H. Okamura, and R. Kageyama, Real-time imaging of the somite segmentation clock:
Revelation of unstable oscillators in the individual presomitic mesoderm cells. Proc. Natl.
Acad. Sci. USA 103, 1313–1318 (2006).
173. J. Dubrulle, M. J. McGrew, and O. Pourquié, FGF signaling controls somite boundary position
and regulates segmentation clock control of spatiotemporal Hox gene activation. Cell 106, 219–
232 (2001).
174. J. Dubrulle and O. Pourquié, fgf8 mRNA decay establishes a gradient that couples axial
elongation to patterning in the vertebrate embryo. Nature 427, 419–422 (2004).
175. A. Goldbeter, D. Gonze, and O. Pourquié, Sharp developmental thresholds defined through
bistability by antagonistic gradients of retinoic acid and FGF signaling. Submitted.
176. R. Diez del Corral and K. G. Storey, Opposing FGF and retinoid pathways: A signalling switch
that controls differentiation and patterning onset in the extending vertebrate body axis.
BioEssays 26, 857–869 (2004).
177. T. S. Gardner, C. R. Cantor, and J. J. Collins, Construction of a genetic toggle switch in
Escherichia coli. Nature 403, 339–342 (2000).
178. C. Garmendia-Torres, A. Goldbeter, and M. Jacquet, Nucleocytoplasmic oscilliations of the
transcription factor Msn2 in yeast are mediated by phosphorylation of its NLS by protein kinase
A. Submitted.
179. E. A. Vitalis, J. L. Costantin, P. S. Tsai, H. Sakakibara, S. Paruthiyil, T. Iiri, J. F. Martini,
M. Taga, A. L. H. Choi, A. C. Charles, and R. I. Weiner, Role of the cAMP signaling pathway in
the regulation of gonadotropin-releasing hormone secretion in GT1 cells. Proc. Natl. Acad. Sci.
USA 97, 1861–1866 (2000).
180. J. Rinzel, A formal classification of bursting mechanisms in excitable systems. Lect. Notes
Biomath 71, 267–281 (1987).
181. L. F. Olsen and H. Degn, Chaos in biological systems. Q. Rev. Biophys. 18, 165–225 (1985).
182. O. Decroly and A. Goldbeter, Birhythmicity, chaos, and other patterns of temporal selforganization in a multiply regulated biochemical system. Proc. Natl. Acad. Sci. USA 79, 6917–
6921 (1982).
183. D. Gonze and A. Goldbeter, Entrainment versus chaos in a model for a circadian oscillator
driven by light–dark cycles. J. Stat. Phys. 101, 649–663 (2000).
184. A. Goldbeter, D. Gonze, G. Houart, J. C. Leloup, J. Halloy, and G. Dupont, From simple to
complex oscillatory behavior in metabolic and genetic control networks. Chaos 11, 247–260
(2001).
185. P. Shen and R. Larter, Chaos in intracellular Ca2þ oscillations in a new model for nonexcitable
cells. Cell Calcium 17, 225–232 (1995).
biological rhythms as temporal dissipative structures
295
186. G. Houart, G. Dupont, and A. Goldbeter, Bursting, chaos and birhythmicity originating from
self-modulation of the inositol 1,4,5-triphosphate signal in a model for intracellular Ca2þ
oscillations. Bull. Math. Biol. 61, 507–530 (1999).
187. U. Kummer, L. F. Olsen, C. J. Dixon, A. K. Green, E. Bornberg-Bauer, and G. Baier,
Switching from simple to complex oscillations in calcium signaling. Biophys. J. 79, 1188–
1195 (2000).
188. K. Nielsen, P. G. Sörensen, and F. Hynne, Chaos in glycolysis. J. Theor. Biol. 186, 303–306
(1997).
189. I. Prigogine, R. Lefever, A. Goldbeter, and M. Herschkowitz-Kaufman, Symmetry-breaking
instabilities in biological systems. Nature 223, 913–916 (1969).
190. J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, Modeling a synthetic multicellular clock:
Repressilators coupled by quorum sensing. Proc. Natl. Acad. Sci. USA 101, 10955–10960
(2004).
191. H. R. Petty, Neutrophil oscillations: Temporal and spatiotemporal aspects of cell behavior.
Immunol. Res. 23, 85–94 (2001).
192. D. Duboule, Time for chronomics? Science 301, 277 (2003).